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For 1 <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> i <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> 4, let xi (<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADFJREFUeNpjYMAE+4HYkoECANK8d3gaVATE/8l1wX6oKywHt0YY2EtpYFEMfhDAKAAAkv8WtNOHnC8AAABQdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIyNjU7PC9tbz48L21hdGg+tOdhZwAAAABJRU5ErkJggg==" class="Wirisformula" alt="greater or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8805;«/mo»«/math»'/> 3) be the number of blanks between ith and (i + 1)th letters. Then, x1 + x2 + x3 + x4 = 15 …..(1) The number of solutions of (1) = coefficient of t15 in (t3 + t4 +….)4 = coefficient of t3 in (1 – t)–4 = coefficient of t3 in [1 + 4C1 + 5C2 t2 + 6C3 t3 + …..] = 6C3 = 20. But 5 letters can be permuted in 5! = 120 ways. Thus, the required number arrangements = (120) (20) = 2400. Hence

Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with at least three between every two. The number of ways in which this can be done is -

Solution for the question - five distinct letters are to be transmitted through a communicationchannel. a total number of 15 blanks is to be inserted between th

Five distinct letters are to be transmitted through a communica-Turito

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Note that 7r (r  N) ends in 7, 9, 3 or 1 (corresponding to r = 1, 2, 3 and 4 respectively). Thus, 7m + 7n cannot end in 5 for any values of m, n  N. In other words, for 7m + 7n to be divisible by 5, it should end in 0. For 7m + 7n to end in 0, the forms of m and n should be as follows : m n 1 4r 4s + 2 2 4r + 1 4s + 3 3 4r + 2 4s 4 4r + 3 4s + 1 Thus, for a given value of m there are just 25 values of n for which 7m + 7n ends in 0. [For instance, if m = 4r, then = 2, 6, 10,….. , 98]  There are 100 × 25 = 2500 ordered pairs (m, n) for which 7m + 7n is divisible by 5. Hence

The number of ordered pairs (m, n), m, n <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAMCAYAAABr5z2BAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAALV/ZLFgAAAGNJREFUeNpjYMANNIB4AhCfA+JvQPwLiP8zEAmCgPgKEGcAsTEQszGQAGShtnIxkAkmAXEAAwXgKRAzUWIALLBwYdq7YAKlYQCKhTNAzEOJIeFI6cAAiFnIMQQ9Jf4hJSUSDQC2pRnou82QIwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjIwODs8L21vPjwvbWF0aD4h58XOAAAAAElFTkSuQmCC" class="Wirisformula" alt="element of" width="16" height="12" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8712;«/mo»«/math»'/> {1, 2, … 100} such that 7m + 7n is divisible by 5 is -

Solution for the question - the number of ordered pairs (m, n), m, n e {1, 2,  100} such that 7m +7n is divisible by 5 is - 50002500

The number of ordered pairs (m, n), m, n e {1, 2,  100} su-Turito

(1) Total number of ways of arranging m things = m!. To find the number of ways in which p particular things are together, we consider p particular things as a group.  Number of ways in which p particular things are together = (m – p + 1)! p! So, number of ways in which p particular things are not together = m! – (m – p + 1)! p! Total number of ways = <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 52 factorial over denominator left parenthesis 13 factorial right parenthesis to the power of 4 end exponent end fraction" width="51" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»52«/mn»«mo»!«/mo»«/mrow»«mrow»«mo»(«/mo»«mn»13«/mn»«mo»!«/mo»«msup»«mrow»«mo»)«/mo»«/mrow»«mrow»«mn»4«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«/math»'/> Hence, both of statements are correct.

Consider the following statements: 1. The number of ways of arranging m different things taken all at a time in which p <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> m particular things are never together is m! – (m – p + 1)! p!. 2. A pack of 52 cards can be divided equally among four players in order in <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 52 factorial over denominator left parenthesis 13 factorial right parenthesis to the power of 4 end exponent end fraction" width="51" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»52«/mn»«mo»!«/mo»«/mrow»«mrow»«mo»(«/mo»«mn»13«/mn»«mo»!«/mo»«msup»«mrow»«mo»)«/mo»«/mrow»«mrow»«mn»4«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«/math»'/>ways. Which of these is/are correct?

Solution for the question - consider the following statements: 1. the number of ways of arranging mdifferent things taken all at a time in which p none of these

Consider the following statements: 1. the number of ways of ar-Turito

Let ‘l’ is associated with ‘r’ , r <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAMCAYAAABr5z2BAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAALV/ZLFgAAAGNJREFUeNpjYMANNIB4AhCfA+JvQPwLiP8zEAmCgPgKEGcAsTEQszGQAGShtnIxkAkmAXEAAwXgKRAzUWIALLBwYdq7YAKlYQCKhTNAzEOJIeFI6cAAiFnIMQQ9Jf4hJSUSDQC2pRnou82QIwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjIwODs8L21vPjwvbWF0aD4h58XOAAAAAElFTkSuQmCC" class="Wirisformula" alt="element of" width="16" height="12" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8712;«/mo»«/math»'/> {1, 2, 3, 4, 5} then ‘2’ can be associated with r, r + 1, ….., 5. Let ‘2’ is associated with ‘j’ then 3 can be associated with j, j + 1, …., 5. Thus required number of functions = <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy sum subscript r equals 1 end subscript superscript 5 end superscript open parentheses not stretchy sum subscript j equals r end subscript superscript 5 end superscript left parenthesis 6 minus j right parenthesis close parentheses" width="150" height="30" style="vertical-align:-9px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»5«/mn»«/mrow»«/msubsup»«mrow»«mfenced separators=¨|¨»«mrow»«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»j«/mi»«mo»=«/mo»«mi»r«/mi»«/mrow»«mrow»«mn»5«/mn»«/mrow»«/msubsup»«mrow»«mo»(«/mo»«mn»6«/mn»«mo»-«/mo»«mi»j«/mi»«mo»)«/mo»«/mrow»«/mrow»«/mrow»«/mfenced»«/mrow»«/mrow»«/math»'/>= <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy sum subscript r equals 1 end subscript superscript 5 end superscript fraction numerator left parenthesis 6 minus r right parenthesis left parenthesis 7 minus r right parenthesis over denominator 2 end fraction" width="146" height="37" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»5«/mn»«/mrow»«/msubsup»«mrow»«mfrac»«mrow»«mo»(«/mo»«mn»6«/mn»«mo»-«/mo»«mi»r«/mi»«mo»)«/mo»«mo»(«/mo»«mn»7«/mn»«mo»-«/mo»«mi»r«/mi»«mo»)«/mo»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mrow»«/math»'/> = <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==" class="Wirisformula" alt="fraction numerator 1 over denominator 2 end fraction" width="18" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/> <img src="data:image/png;base64,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" class="Wirisformula" alt="open parentheses not stretchy sum subscript r equals 1 end subscript superscript 5 end superscript left parenthesis 42 minus 13 r plus r to the power of 2 end exponent right parenthesis close parentheses" width="170" height="28" style="vertical-align:-7px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced separators=¨|¨»«mrow»«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»5«/mn»«/mrow»«/msubsup»«mrow»«mo»(«/mo»«mn»42«/mn»«mo»-«/mo»«mn»13«/mn»«mi»r«/mi»«mo»+«/mo»«msup»«mrow»«mi»r«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«mo»)«/mo»«/mrow»«/mrow»«/mrow»«/mfenced»«/math»'/> = <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==" class="Wirisformula" alt="fraction numerator 1 over denominator 2 end fraction" width="18" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«mn»1«/mn»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/math»'/> <img src="data:image/png;base64,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" class="Wirisformula" alt="open parentheses 42.5 minus 13. fraction numerator 6.5 over denominator 2 end fraction plus fraction numerator 5.6.11 over denominator 6 end fraction close parentheses" width="193" height="38" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced separators=¨|¨»«mrow»«mn»42.5«/mn»«mo»-«/mo»«mn»13«/mn»«mo».«/mo»«mfrac»«mrow»«mn»6.5«/mn»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mn»5.6«/mn»«mo».«/mo»«mn»11«/mn»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/>= 35 Hence (a) is correct answer.

The total number of function ‘ƒ’ from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that ƒ(i) <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> ƒ(j), <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAOCAYAAAD0f5bSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAANvpXuIwAAAIlJREFUeNpjYECAK0CswoAdgMQvYZNoBuIiHJqKoPIYwBSID+LQtB+IzXHIMTwFYkE0MT4gfs6AB6wB4v9Y8Bp8mnyAeBWa2CqoOE7AAsSfoDSM/wWImRgIgOVA7IfHZqwgGohnQ9mzoXyCADm0sIUmTgCKlxw88YYzBTzFk0KwAiVo/KgwUAsAABDxHJ8Oxq5sAAAAZXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaSBtYXRodmFyaWFudD0ibm9ybWFsIj4mI3gyMjAwOzwvbWk+PC9tYXRoPiVDCQMAAAAASUVORK5CYII=" class="Wirisformula" alt="straight for all" width="13" height="14" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#8704;«/mi»«/math»'/> i &lt; j, is equal to-

Solution for the question - the total number of function  from the set {1, 2, 3} into the set{1, 2, 3, 4, 5} such that (i) 6050

The total number of function  from the set {1, 2, 3} in-Turito

|x| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAGF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8gc3RyZXRjaHk9ImZhbHNlIj4mI3gyMUQyOzwvbW8+PC9tYXRoPkdgprsAAAAASUVORK5CYII=" class="Wirisformula" alt="not stretchy rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo stretchy=¨false¨»§#8658;«/mo»«/math»'/> –k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> x <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k ….(1) &amp; |y| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAGF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8gc3RyZXRjaHk9ImZhbHNlIj4mI3gyMUQyOzwvbW8+PC9tYXRoPkdgprsAAAAASUVORK5CYII=" class="Wirisformula" alt="not stretchy rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo stretchy=¨false¨»§#8658;«/mo»«/math»'/> –k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> y <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k ….(2) &amp; |x – y| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAGF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8gc3RyZXRjaHk9ImZhbHNlIj4mI3gyMUQyOzwvbW8+PC9tYXRoPkdgprsAAAAASUVORK5CYII=" class="Wirisformula" alt="not stretchy rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo stretchy=¨false¨»§#8658;«/mo»«/math»'/> |y – x| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k ….(3) <img src="https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/generalfeedback/486227/1/924158/Picture18.png" alt="" width="199" height="223" /> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAGF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8gc3RyZXRjaHk9ImZhbHNlIj4mI3gyMUQyOzwvbW8+PC9tYXRoPkdgprsAAAAASUVORK5CYII=" class="Wirisformula" alt="not stretchy rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo stretchy=¨false¨»§#8658;«/mo»«/math»'/> – k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> y – x <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAKCAYAAAC0VX7mAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAADlJREFUeNpjYCANlAAxHwMVQTUQPwRia1I0/ScStwIxE7VcOhdq6FJqurCDGi4kKwwJxTIPwyhABgA3dxtvdcXJJQAAAGF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8gc3RyZXRjaHk9ImZhbHNlIj4mI3gyMUQyOzwvbW8+PC9tYXRoPkdgprsAAAAASUVORK5CYII=" class="Wirisformula" alt="not stretchy rightwards double arrow" width="20" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo stretchy=¨false¨»§#8658;«/mo»«/math»'/> x – k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> y <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> x + k <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAKCAYAAABSfLWiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAACBJREFUeNpjYCAM/kMxRYAqhowC/AEZAJVbThdDBgYAAO75C+VyngYIAAAAUHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4mI3gyMjM0OzwvbW8+PC9tYXRoPhpvPpoAAAAASUVORK5CYII=" class="Wirisformula" alt="therefore" width="17" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8756;«/mo»«/math»'/>Number of points having integral coordinates = (2k + 1)2 – 2[k + (k – 1) + …. + 2 + 1] = (3k2 + 3k + 1).

The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x|  <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k, |y| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k, |x – y| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> k ; is-

Solution for the question - the number of points in the cartesian plane with integral co-ordinatessatisfying the inequalities |x| none of these

The number of points in the cartesian plane with integral co-or-Turito

The required no. of ways = no. of solution of the equation (x1 + x2 + x3 + x4 + x5 + x6 = K) Where 0  xi 9, i = 1, 2, …6, where 0 &lt; K &lt; 18 = Coefficient of xK in (1 + x + x2 +….. + x9)6 = Coefficient of xK in <img src="data:image/png;base64,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" class="Wirisformula" alt="open parentheses fraction numerator 1 minus x to the power of 10 end exponent over denominator 1 minus x end fraction close parentheses to the power of 6 end exponent" width="84" height="44" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msup»«mrow»«mfenced separators=¨|¨»«mrow»«mfrac»«mrow»«mn»1«/mn»«mo»-«/mo»«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»10«/mn»«/mrow»«/msup»«/mrow»«mrow»«mn»1«/mn»«mo»-«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mfenced»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msup»«/math»'/> = Coefficient of xk in (1 – 6x10 + 15 x20 – ….) (1 + 6 C1x + 7 C2 x2 + …. +(7 – K – 10 – 1) CK–10 xK–10 + ….+(7 + K – 1) CK xK + …) = k + 6CK – 6. K–4CK–10 = k + 6C6 – 6. K–4C6 .

The numbers of integers between 1 and 106 have the sum of their digit equal to K(where 0 &lt; K &lt; 18) is -

Solution for the question - the numbers of integers between 1 and 106 have the sum of their digit equalto k(where 0 < k < 18) is - k + 6c6 6 k 4c6

The numbers of integers between 1 and 106 have the sum of their-Turito

Total number of points = m +n + k. Therefore the total number of triangles formed by these points is m + n + kC3. But out of these m + n + k points, m points lie on I1, n points lie on I2 and k points lie on I3 and by joining three points on the same line we do not obtain a triangle. Hence the total number of triangles is m + n + kC3 – mC3 – nC3 – kC3.

The straight lines I1, I2, I3 are parallel and lie in the same plane. A total number of m points are taken on I1 ; n points on I2 , k points on I3. The maximum number of triangles formed with vertices at these points are -

Solution for the question - the straight lines i1, i2, i3 are parallel and lie in the same plane. atotal number of m points are taken on i1 ; n points on i2 , k

The straight lines i1, i2, i3 are parallel and lie in the same -Turito

If the line <img src="data:image/png;base64,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" class="Wirisformula" alt="l x plus m y equals 1" width="84" height="15" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»l«/mi»«mi»x«/mi»«mo»+«/mo»«mi»m«/mi»«mi»y«/mi»«mo»=«/mo»«mn»1«/mn»«/math»'/> is a normal to the hyperbola <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGIAAAAqCAYAAABIro4PAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAAAo9JREFUeNrtW0FEw1EYf3bIJJFMskNMkqRr0qHLDpnMbtkhmejQabp0yCSJdNghXTokScbMDkmXJEm6JJlO0TGZbskksb5PX/z792Jb1vu+5/348f+/XX5v3/fe972935T6O6rEN+AFsFfxhSStDSMAnANeO6088Oq0mscY8MxpNYtu2ncjTqs59AEPaYJOq8HsOgK2O63NwQxwTTO+Qp99ASfWz0DvNnBSM54GrjLTWjewvevyvKeAm7/05l6awDRw3TfWCrwDdjLTWjfGgVl6jgJPGGuNAfO+sSVgxpYacEyt3o0nszgCtd163kPAe2DQlkCk6SeBIQFaK57nLGm3AqPAAk0qKUDvOZ0NIrQaAjYEASdzQAWvDXjFfGtC7ALjwH3glA1BCFGr5/3iJ4B7ArZRbGNLNgShBVgEhjWf5aiT4ooEtaRxxqf6DDU+VgMDcMlYH+4os5LOL40Ct9MRATqtDsSg+vwxT7lA2I9qDXSBEBbQP0W7UTYjw0zodVuT25oc3IpwgXAQGQhp7jmTev+lMZDmnrPe7ffq9JqHNPeclW4/Du65iqr9Vs5Ktx8H9xwW3ZIgvXUXtGVgmboMvNDo0WQWB/ccXgrlalwJ4tx+mDUbwA4KSp4m7AUX9xx6mBYpcR6pCJ9qsl6c2y+pfhq28HI+ZqI3rgGoFZ0bC57EyWpWiTi3H2aT/7brTLM1cQFaKwd8Y7j9PEsvwP4OBJ9fGNeyimY8aEMgnnzvo4xPocO0gv3AFX0iPRAP6rtvFG0gBcb1bEczPq8sMCHj/yC2aNljXdinjoMjsLNL+cZa6FxhxT+DFqkbydDqKGvaVw4oUrGO0nuYxlLKUoSY6sIEiVMw3ungmZD0xX4ABTHtFp4NjSkAAAD/dEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtc3VwPjxtaT5hPC9taT48bW4+MjwvbW4+PC9tc3VwPjwvbWZyYWM+PG1vPi08L21vPjxtZnJhYz48bXN1cD48bWk+eTwvbWk+PG1uPjI8L21uPjwvbXN1cD48bXN1cD48bWk+YjwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21mcmFjPjxtbz49PC9tbz48bW4+MTwvbW4+PC9tYXRoPp+KhTMAAAAASUVORK5CYII=" class="Wirisformula" alt="fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1" width="98" height="42" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«msup»«mrow»«mi»y«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»b«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mn»1«/mn»«/math»'/> then <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator a to the power of 2 end exponent over denominator l to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator m to the power of 2 end exponent end fraction equals" width="92" height="42" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«msup»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»l«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«msup»«mrow»«mi»b«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»m«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«/math»'/>

Solution for the question - if the line (x+my=1 is a normal to the hyperbola(x^(2))/(a^(2))-(y^(2))/(b^(2))=1 then (a^(2))/(p^(2))-(b^(2))/(m^(2))= (a^(2)-b^(2))^(2) '/>

If the line (x+my=1 is a normal to the hyperbola (x^(2))/(a^(2)-Turito

If the tangents drawn from a point on the hyperbola <img src="data:image/png;base64,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" class="Wirisformula" alt="x to the power of 2 end exponent minus y to the power of 2 end exponent equals a to the power of 2 end exponent minus b to the power of 2 end exponent" width="124" height="19" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«mo»-«/mo»«msup»«mrow»«mi»y«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«mo»=«/mo»«msup»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«mo»-«/mo»«msup»«mrow»«mi»b«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/math»'/> to the ellipse <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1" width="98" height="42" style="vertical-align:-15px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfrac»«mrow»«msup»«mrow»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»a«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«msup»«mrow»«mi»y«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«mrow»«msup»«mrow»«mi»b«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mn»1«/mn»«/math»'/> make angles α and β with the transverse axis of the hyperbola, then

Solution for the question - if the tangents drawn from a point on the hyperbola x^(2)-y^(2)=a^(2)-b^(2)to the ellipse (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 make ang

If the tangents drawn from a point on the hyperbola x^(2)-y^(2)-Turito

The eccentricity of the hyperbola whose latus rectum subtends a right angle at centre is

Solution for the question - the eccentricity of the hyperbola whose latus rectum subtends a right angleat centre is 2cot 18^(@)   2tan 18^(@)   2cos 36^(@)

The eccentricity of the hyperbola whose latus rectum subtends a-Turito

An ordered pair is made up of the ordinate and the abscissa of the x coordinate, with two values given in parenthesis in a certain sequence. Now we have given the LCM as: r2t4s2 Consider following cases: Case 1: if p contains r2 then q will have rk, for the value k=0,1. So 2 ways. Case 2: if q contains r2 then p will have rk, for the value k=0,1. So 2 ways. Case 3:Both p and q contain r2 So 1way. So after this we can say that: exponent of r=2+2+1 = 5 ways. Similarly exponent of t=4+4+1=9 ways. exponent of s=2+2+1 = 5 ways. So total ways will be: 5 x 5 x 9 = 225 ways.

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r2t4s2, then the number of ordered pair (p, q) is –

Solution for the question - if r, s, t are prime numbers and p, q are the positive integers such thatthe lcm of p, q is r2t4s2, then the number of ordered pair

If r, s, t are prime numbers and p, q are the positive integers-Turito

Now we have given that a rectangle has sides of (2m – 1) &amp; (2n – 1) units as shown in the figure composed of squares having an edge length of one unit. There are 2n horizontal lines and 1 2 m vertical lines (numbered 1,2.......2n). Two horizontal lines, one with an even number and one with an odd number, as well as two vertical lines must be chosen in order to create the necessary rectangle. Then the number of rectangles will be: (1+3+5+......+(2m−1))(1+3+5+......+(2n−1))=m2n2 So it is m2n2.

A rectangle has sides of (2m – 1) &amp; (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length <img 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" alt="" />

Solution for the question - a rectangle has sides of (2m  1) & (2n  1) units as shown in thefigure composed of squares having edge length one unit then no. ofre

A rectangle has sides of (2m  1) & (2n  1) units as shown-Turito

nCr + 2nCr+1 + nCr+2 is equal to (2  <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/>r <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAALCAYAAAB24g05AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAKIPF7gAAAADlJREFUeNpjYCAeWALxfgYyAEzjXiibfhr3k6oRBv4DcREDBYBiF9DMIFBgmlNq0G4GeoAfBDAKAACLixa2LHq2kwAAAFB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4MjI2NDs8L21vPjwvbWF0aD4p6IARAAAAAElFTkSuQmCC" class="Wirisformula" alt="less or equal than" width="16" height="11" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8804;«/mo»«/math»'/> n)

Solution for the question - ncr + 2ncr+1 + ncr+2 is equal to (2 n+1cr+1

Ncr + 2ncr+1 + ncr+2 is equal to (2 -Turito

The coefficient of <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAPCAYAAAAGRPQsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAOJ5y/mQAAALdJREFUeNpjYCAP3ABiHyA+A8S/gLicTHMYmID4DxAvBmJpIDYA4nfkGmYOdREMWAPxXnINiwTihWj8+eQaNgmIM9D4yeiKQAIdWDQ3oyleB8ReSPwNaHw4OAfE4kj8RCCegqYGFNgcePhw4AHEfVC2CyUBCwO7gdgeiC8AsTClhhVAE6IepQaB0swaqFcjKTFICYg3ATEXEPNAEyZZ3hQF4m1omn2gWYYkwAZNO9JY5JZDY5j+AABdeyAPgdWJqgAAAGB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXN1cD48bWk+eDwvbWk+PG1pPm48L21pPjwvbXN1cD48L21hdGg+PrUppgAAAABJRU5ErkJggg==" class="Wirisformula" alt="x to the power of n" width="19" height="15" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«/math»'/> in <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator x plus 1 over denominator left parenthesis x minus 1 right parenthesis squared left parenthesis x minus 2 right parenthesis end fraction" width="112" height="39" style="vertical-align:-16px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«mrow»«mo»(«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«msup»«mo»)«/mo»«mn»2«/mn»«/msup»«mo»(«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mo»)«/mo»«/mrow»«/mfrac»«/math»'/> is

Solution for the question - the coefficient of x^(n) in (x+1)/((x-1)^(2)(x-2)) is 1+2n-(3)/(2^(n-1)) 1+2n-(3)/(2^(n-1)) 1-2n-(3)/(2^(n-1)) 1-2n-(3)/(2^(n+1))

The coefficient of x^(n) in (x+1)/((x-1)^(2)(x-2)) is -Turito

A permutation is the non-replaceable selection of r items from a set of n items in which the order is important. <img src="data:image/png;base64,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" class="Wirisformula" alt="P presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial end fraction" width="104" height="39" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»P«/mi»«mi»r«/mi»«none/»«mprescripts/»«none/»«mi»n«/mi»«/mmultiscripts»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mo»(«/mo»«mi»n«/mi»«mo»!«/mo»«mo»)«/mo»«mo»§#160;«/mo»«/mrow»«mrow»«mo»(«/mo»«mi»n«/mi»«mo»-«/mo»«mi»r«/mi»«mo»)«/mo»«mo»!«/mo»«/mrow»«/mfrac»«/math»'/> A combination is created by selecting r items from a group of n items without replacing them and without regard to their order. <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript n subscript r equals space fraction numerator left parenthesis n factorial right parenthesis space over denominator left parenthesis n minus r right parenthesis factorial cross times r factorial end fraction" width="136" height="39" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«none/»«mi»n«/mi»«/mmultiscripts»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mo»(«/mo»«mi»n«/mi»«mo»!«/mo»«mo»)«/mo»«mo»§#160;«/mo»«/mrow»«mrow»«mo»(«/mo»«mi»n«/mi»«mo»-«/mo»«mi»r«/mi»«mo»)«/mo»«mo»!«/mo»«mo»§#215;«/mo»«mi»r«/mi»«mo»!«/mo»«/mrow»«/mfrac»«/math»'/> Here we have given the word as: MISSISSIPPI. Now different words can be formed by jumbling the letters of this word in which not two S are adjacent, those can be found out as follows: Here, I = 4 times, S= 4 times, P = 2 times, M= 1 time So the number of words will be: <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 8 subscript 4 cross times space fraction numerator left parenthesis 7 factorial right parenthesis space over denominator left parenthesis 4 right parenthesis factorial cross times 2 factorial end fraction
equals 7 cross times space C presuperscript 6 subscript 4 cross times C presuperscript 8 subscript 4" width="121" height="69" style="vertical-align:-35px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«none/»«mn»8«/mn»«/mmultiscripts»«mo»§#215;«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mo»(«/mo»«mn»7«/mn»«mo»!«/mo»«mo»)«/mo»«mo»§#160;«/mo»«/mrow»«mrow»«mo»(«/mo»«mn»4«/mn»«mo»)«/mo»«mo»!«/mo»«mo»§#215;«/mo»«mn»2«/mn»«mo»!«/mo»«/mrow»«/mfrac»«mspace linebreak=¨newline¨/»«mo»=«/mo»«mn»7«/mn»«mo»§#215;«/mo»«mo»§#160;«/mo»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«none/»«mn»6«/mn»«/mmultiscripts»«mo»§#215;«/mo»«mmultiscripts»«mi»C«/mi»«mn»4«/mn»«none/»«mprescripts/»«none/»«mn»8«/mn»«/mmultiscripts»«/math»'/>

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?

Solution for the question - how many different words can be formed by jumbling the letters in the wordmississippi in which not two s are adjacent ? 8. 6c4 . 7c4

How many different words can be formed by jumbling the letters -Turito

The value of 50C4 + <img src="data:image/png;base64,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" class="Wirisformula" alt="not stretchy sum subscript r equals 1 end subscript superscript 6 end superscript 56 minus r C subscript 3 end subscript" width="112" height="28" style="vertical-align:-7px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mrow»«msubsup»«mo stretchy=¨false¨»∑«/mo»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«mrow»«mn»6«/mn»«/mrow»«/msubsup»«mrow»«mn»56«/mn»«mo»-«/mo»«mi»r«/mi»«msub»«mrow»«mi»C«/mi»«/mrow»«mrow»«mn»3«/mn»«/mrow»«/msub»«/mrow»«/mrow»«/math»'/>is -

Solution for the question - the value of 50c4 + sum_(r=1)^(6)56-rc_(3) is - 56c4