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Matches whose prediction are correct can be selected in <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 20 subscript 10" width="44" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»10«/mn»«none/»«mprescripts/»«none/»«mn»20«/mn»«/mmultiscripts»«/math»'/> ways. Since each match can result either in a win, loss or tie for the home team. Now, each wrong prediction can be made in two ways (i.e. the correct result is win and the person predicts either lose or tie) and there are 10 matches for which he predicted wrong. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct = <img src="data:image/png;base64,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" class="Wirisformula" alt="N u m b e r space o f space c o r r e c t space p r e d i c t i o n s space cross times space N u m b e r space o f space i n c o r r e c t space p r e d i c t i o n s" width="516" height="16" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»N«/mi»«mi»u«/mi»«mi»m«/mi»«mi»b«/mi»«mi»e«/mi»«mi»r«/mi»«mo»§#160;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»c«/mi»«mi»o«/mi»«mi»r«/mi»«mi»r«/mi»«mi»e«/mi»«mi»c«/mi»«mi»t«/mi»«mo»§#160;«/mo»«mi»p«/mi»«mi»r«/mi»«mi»e«/mi»«mi»d«/mi»«mi»i«/mi»«mi»c«/mi»«mi»t«/mi»«mi»i«/mi»«mi»o«/mi»«mi»n«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mo»§#215;«/mo»«mo»§#160;«/mo»«mi»N«/mi»«mi»u«/mi»«mi»m«/mi»«mi»b«/mi»«mi»e«/mi»«mi»r«/mi»«mo»§#160;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»i«/mi»«mi»n«/mi»«mi»c«/mi»«mi»o«/mi»«mi»r«/mi»«mi»r«/mi»«mi»e«/mi»«mi»c«/mi»«mi»t«/mi»«mo»§#160;«/mo»«mi»p«/mi»«mi»r«/mi»«mi»e«/mi»«mi»d«/mi»«mi»i«/mi»«mi»c«/mi»«mi»t«/mi»«mi»i«/mi»«mi»o«/mi»«mi»n«/mi»«mi»s«/mi»«/math»'/> <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 20 subscript 10 cross times space 2 to the power of 10" width="91" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»10«/mn»«none/»«mprescripts/»«none/»«mn»20«/mn»«/mmultiscripts»«mo»§#215;«/mo»«mo»§#160;«/mo»«msup»«mn»2«/mn»«mn»10«/mn»«/msup»«/math»'/> Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 20 subscript 10 cross times space 2 to the power of 10" width="91" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»10«/mn»«none/»«mprescripts/»«none/»«mn»20«/mn»«/mmultiscripts»«mo»§#215;«/mo»«mo»§#160;«/mo»«msup»«mn»2«/mn»«mn»10«/mn»«/msup»«/math»'/>

A person predicts the outcome of 20 cricket matches of his home team. Each match can result either in a win, loss or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to :

Solution for the question - a person predicts the outcome of 20 cricket matches of his home team. eachmatch can result either in a win, loss or tie for the home

A person predicts the outcome of 20 cricket matches of his home-Turito

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The foot of the perpendicular from the point <img src="data:image/png;base64,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" class="Wirisformula" alt="left parenthesis 3 , 3 pi divided by 4 right parenthesis" width="69" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mo»(«/mo»«mn»3,3«/mn»«mi»π«/mi»«mo»/«/mo»«mn»4«/mn»«mo»)«/mo»«/math»'/> on the line <img src="data:image/png;base64,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" class="Wirisformula" alt="r left parenthesis c o s space theta minus s i n space theta right parenthesis equals 6 square root of 2" width="169" height="19" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mo»(«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»-«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mo»=«/mo»«mn»6«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/math»'/> is

Solution for the question - the foot of the perpendicular from the point (3,3pi//4) on the line r(costheta-sin theta)=6sqrt2 is (3,2pi//4)   (-3,pi//2)

The foot of the perpendicular from the point (3,3pi//4) on the -Turito

The point of intersection of the lines <img src="data:image/png;base64,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" class="Wirisformula" alt="2 c o s space theta plus s i n space theta equals 1 over r comma c o s space theta plus s i n space theta equals 1 over r" width="281" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mi»r«/mi»«/mfrac»«mo»,«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mi»r«/mi»«/mfrac»«/math»'/> is

Solution for the question - the point of intersection of the lines 2cos theta+sin theta=(1)/(r),costheta+sin theta=(1)/(r) is (2,(pi)/(2))   (2,(pi)/(4))

The point of intersection of the lines 2cos theta+sin theta=(1)-Turito

The line passing through <img src="data:image/png;base64,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" class="Wirisformula" alt="open parentheses negative 1 comma fraction numerator pi over denominator 2 end fraction close parentheses" width="63" height="38" style="vertical-align:-14px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mfenced separators=¨|¨»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mfrac»«mrow»«mi»π«/mi»«/mrow»«mrow»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mfenced»«/math»'/> and perpendicular to <img src="data:image/png;base64,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" class="Wirisformula" alt="square root of 3 s i n space theta plus 2 c o s space theta equals 4 over r" width="169" height="35" style="vertical-align:-12px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»3«/mn»«/msqrt»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«mn»2«/mn»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»=«/mo»«mfrac»«mn»4«/mn»«mi»r«/mi»«/mfrac»«/math»'/> is

Solution for the question - the line passing through (-1,(pi)/(2)) and perpendicular to sqrt3sintheta+2cos theta=(4)/(pi) is 5=2sqrt3r sin theta+4r cos theta

The line passing through (-1,(pi)/(2)) and perpendicular to sqr-Turito

The equation of the line passing through <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIwAAAARCAYAAAAPOESjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAOJ5y/mQAAArpJREFUeNrdWc9HZVEcv57xJGNo8STJkGS0SCRJMpvMIi3ebqRVhidJ2xZpNYa0TiTJaBEZIxkjkiRPIklmMSKt+geSPNejviefy+l1zj3fe6/z7rnvw2dx3v3h+/m8c873+z3X815jhrjkNR76iOUYzy3Bkyhw0UMr+nuJp15jYp04GfPZMrzhwFUPregvYybaQDdxkXiZglkF4i0xp7k+QNwh3hMrxD3iZ+l6P/Ekgrl9GdE/TPwF3T5iU00qpf64WxYXW8QS8SkFwxbwZ6lQwgTpxvgDTKs16ATGxd32XdR/TJwgvse4B/FPKO59o3+ZOFuH4OttWA6rq6C4Jgw6Z75njlGXcDx0Sb8KH4lXHP37xBFHJ4wouh7xrI7zmmfFatnQXFvTrCYVhoiHhns4HrqkX4cKR7/IY3lHJ8xPYgfybVH6XYzHDc+GpZL/xDZmDHl4FAaOhy7p1y2MMkd/tU5bZJIt+QbGBRBbbWuCVrKC2uU3VrCPFKXrJnxDfNWM6a9FE/EMxbBRf5Uh1ESbE+Yd/lR5/JCwlRSxXBDH8L4czLrWnD2kOWFs6JfRQtwlfgm5x89KSgq2yqMINYWplQ40t2tW5p1DKcmW/gCdmCxdUVLyQchW5MKEEcXbpjT+itwep5UO8DfEUD9G0cvx0CX9Ap+wEzVHLfpdb6tXiVPS+DtxJWErOQPjVSae1fw2C4/Saqtt6Bf1zw7Smwlv9Ns+uDMZto1rxZCCTz5B/YEV3aVYHdxWMo/Dq2+SaYM4hxg1dBuqeDkeuqT/DxYHB8puS3wD6bU4UcKK5DDDiihOZQwgp24lbCUL2JIfULQeK85SVEfjunh1Hrqon9vAaD+NpP3hbBsdi2vQfXxTxZvEw6zpf8G0l86nebHd/mPm03pC5O1SxHjjeJg1/amj3XAI1ejxZkb/MywU/r1nzLpAAAAA63RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4oPC9tbz48bW8+LTwvbW8+PG1uPjE8L21uPjxtbz4sPC9tbz48bWk+JiN4M0MwOzwvbWk+PG1vPi88L21vPjxtbj42PC9tbj48bW8+KTwvbW8+PG1vPiw8L21vPjxtbz4oPC9tbz48bW4+MTwvbW4+PG1vPiw8L21vPjxtaT4mI3gzQzA7PC9taT48bW8+LzwvbW8+PG1uPjI8L21uPjxtbz4pPC9tbz48L21hdGg+Draz3AAAAABJRU5ErkJggg==" class="Wirisformula" alt="left parenthesis negative 1 comma pi divided by 6 right parenthesis comma left parenthesis 1 comma pi divided by 2 right parenthesis" width="140" height="17" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨ »«mo»(«/mo»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mi»π«/mi»«mo»/«/mo»«mn»6«/mn»«mo»)«/mo»«mo»,«/mo»«mo»(«/mo»«mn»1«/mn»«mo»,«/mo»«mi»π«/mi»«mo»/«/mo»«mn»2«/mn»«mo»)«/mo»«/math»'/> is

Solution for the question - the equation of the line passing through (-1,pi//6),(1,pi//2) is r(2sin theta-4cos theta)=6   r(-3sin theta+4cos theta)=6

The equation of the line passing through (-1,pi//6),(1,pi//2) i-Turito

The length of the perpendicular from (-1, π/6) to the line <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALMAAAATCAYAAADbCDwdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAABCxJREFUeNrtWl1IFUEUXuQiEiKJhYQFET74ECKYmFSEIBESEmFQRIQYEiEiESgY1EskEhT1aBG9FETZS5gQISIhRthFQuRGRA/Rg2AiUSL2c4a+hWmY3/3z5t0PPnR3dnfOfOfszJmz1/NSJIFS4u8QTJEib3CCOJ7KkF84TxxMZXDGc+LZAhjnIGIk71FLnErj0hlbiF/xtxDwCrGS90bWccfNxFHid+IKMUe8SawI0Qd7VlGENu8jjhA/EyeIjeug2zniU8n5OPRLAiZN64mTMfQbmV51CGYeLAc8RiwWrhsLaGw1cTbCwTdCdP/l2Er8sA7OZzq1K85HqV8SsNV0EkEdtY6R6DVE7La8djlPhM8Sq4RzYw5LYBRVhJ3EBWKJwz3LeRzMtpr2JLi3kup1mdiHZfENpvI+zuADFg/eRZxXtA0QPxHXECh9iv7544vESuI9LC+LEMqEFtwj4iGxNcFg7icOO1yv089HKSYXtsyvYqYs59rZbHmX+A0+vK94mUz+CKNpE/FlAoGs1OsRBLqOi8ToL9Y8lAVcB/KYVoVww4ZnsP6PCsc9EPAglrYqOGiTYZDDWJJEMMcfSjCY3xIPW1xn0o8P5BniVWIZ9GQv/BG0lyNV64BeZQi2oQD+CKNpsWaFiaKubtQrpzDWw9trY1ivZvPYYjAwJyxh7LhLct2iRdI/pxGqMoZgLpWc2038QswY+rDRjy993TKkgxcks1cugD/Caroaw0xspVcRlnHPMZg9bkZgs+o0ZlERxzGjtCnuF/svwjIpImOw07932eF8mBmjGisHa9svtF0j3rB0kkk/3/4FIaWQtYspBTtecvRHFJrGEcxWejV6+i9UpjSDn6GmFW3bsMsdl8xkYv8qe5o885e0SkW+1kB87DgLmHAJwczqyK+Fto/EPY5O0unXYCh5qUpiKi11/giradxphlavk4rk3scL72990QYrmrYS5HSnDf2r7DHZyZePZBvcjoiD2Qf76vWLW273Et8bUgxX/VoV4/LRhn2GiF7k2C7+CKtpUhtAqV4sD+uMoDRXK8nPRLCd9ilD/yp7THb6b2tWsizNOG54XDeALP15gP9vE68EcIxOv3pDpYNtNEclaQDLdWsc/RFW027JpjMOSPUaMeyiZR9NJpB7ZSBaM5bWM5rnNEDcCkP/KntMdvp4x5XwmCOnuB1/XMF8h/gDeiwYAiioflnMshlUKnq5FTODapQ/ziroNRDAH2E1jeOjibVei565sD/l/VscZ3XnZ5jmV5B7yQa3hMBYwzU1igpFiYU9Nnb6b2wWm5BZh41OGGwm/iQ+8ex+w2KrH48dcOoaxtUpeeYcxj2vSAFs/BFG07g+ZwfRSzuY9IdGekxaltg2Mv6LHxoxsK+D6U9A9cs2C+btBTp+lid3pWGwcdCeSrA++ANq/WgRUQjTygAAARh0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+cjwvbWk+PG1vPig8L21vPjxtbj4zPC9tbj48bWk+czwvbWk+PG1pPmk8L21pPjxtaT5uPC9taT48bW8+JiN4QTA7PC9tbz48bWk+JiN4M0I4OzwvbWk+PG1vPis8L21vPjxtc3FydD48bW4+MzwvbW4+PC9tc3FydD48bWk+YzwvbWk+PG1pPm88L21pPjxtaT5zPC9taT48bW8+JiN4QTA7PC9tbz48bWk+JiN4M0I4OzwvbWk+PG1vPik8L21vPjxtbz49PC9tbz48bW4+MzwvbW4+PC9tYXRoPjv1Cy8AAAAASUVORK5CYII=" class="Wirisformula" alt="r left parenthesis 3 s i n space theta plus square root of 3 c o s space theta right parenthesis equals 3" width="179" height="19" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mo»(«/mo»«mn»3«/mn»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«msqrt»«mn»3«/mn»«/msqrt»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mo»=«/mo»«mn»3«/mn»«/math»'/> is

Solution for the question - the length of the perpendicular from (-1, /6) to the line r(3sintheta+sqrt3cos theta)=3 is sqrt2   sqrt3  3

The length of the perpendicular from (-1, /6) to the line r(3-Turito

Complete step-by-step answer: According to the problem, we need to find the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time. Let us first fix a number in a unit place and find the total number of words possible due on fixing this number. We need to arrange the remaining three places with three digits. We know that the number of ways of arranging n objects in n places is n! ways. So, we get <span id="MathJax-Element-2-Frame" class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;">3!=6 numbers on fixing the unit place with a particular digit. Now, let us find the sum of all digits. We get sum as <span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/math&gt;">2+3+4+5=14.<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAFCAYAAAB8ZH1oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYBhGAAAAzQABhUzAtQAAADl0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIi8+Llve3gAAAABJRU5ErkJggg==" class="Wirisformula" alt="&amp;lt;br&amp;gt; &amp;lt;/br&amp;gt;" width="10" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»«br /»«/mn»«/math»'/> <span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/math&gt;">Now, we get a sum of digits in units place for all the numbers as <span id="MathJax-Element-4-Frame" class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;">14×6=84.<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAFCAYAAAB8ZH1oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYBhGAAAAzQABhUzAtQAAADl0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIi8+Llve3gAAAABJRU5ErkJggg==" class="Wirisformula" alt="&amp;lt;br&amp;gt; &amp;lt;/br&amp;gt;" width="10" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»«br /»«/mn»«/math»'/> <span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/math&gt;"><span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;">We use the same digits in ten, hundred and thousand places also. So, the sum of those digits will also be 84 but with the multiplication of its place value. i.e., We multiply the sum of the digits in tenth place with 10, hundredth place with 100 and so on. We then add these sums. So, we get the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time is <span id="MathJax-Element-5-Frame" class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;">(84×1000)+(84×100)+(84×10)+(84×1)<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAFCAYAAAB8ZH1oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYBhGAAAAzQABhUzAtQAAADl0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIi8+Llve3gAAAABJRU5ErkJggg==" class="Wirisformula" alt="Error converting from MathML to accessible text." width="10" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mrow»«mo»«br /»«/mo»«/mrow»«/math»'/> <span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;">Sum =  <span id="MathJax-Element-7-Frame" class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;84000&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;8400&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;840&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;">84000+8400+840+84<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAFCAYAAAB8ZH1oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYBhGAAAAzQABhUzAtQAAADl0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIi8+Llve3gAAAABJRU5ErkJggg==" class="Wirisformula" alt="&amp;lt;br&amp;gt; &amp;lt;/br&amp;gt;" width="10" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»«br /»«/mn»«/math»'/> <span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;"><span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;84000&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;8400&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;840&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;">Sum = 93324 <span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;"><span class="MathJax" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;84000&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;8400&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;840&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;">We have found the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time as 93324. <span class="MathJax" style="-webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline-table; line-height: 49px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #332e2b; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;"><span class="MJX_Assistive_MathML" style="-webkit-tap-highlight-color: rgba(255, 255, 255, 0); top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: static; padding: 0px; border: 0px; display: inline; transition: none 0s ease 0s; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; width: 1px !important; overflow: hidden !important;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAQAAAAFCAYAAABirU3bAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYKABAAAAVQAB8vOvGAAAAE50RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4QTA7PC9tbz48L21hdGg+SHplFQAAAABJRU5ErkJggg==" class="Wirisformula" alt="Error converting from MathML to accessible text." width="4" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mrow»«mo» «/mo»«/mrow»«/math»'/> <span class="MathJax" style="-webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; line-height: 49px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: #332e2b; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;84000&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;8400&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;840&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;84&lt;/mn&gt;&lt;/math&gt;"><span class="MJX_Assistive_MathML" style="-webkit-tap-highlight-color: rgba(255, 255, 255, 0); top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: static; padding: 0px; border: 0px; display: inline; transition: none 0s ease 0s; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; width: 1px !important; overflow: hidden !important;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAQAAAAFCAYAAABirU3bAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAEx0lWhwAAAAxJREFUeNpjYKABAAAAVQAB8vOvGAAAAE50RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW4+JiN4QTA7PC9tbj48L21hdGg+OmZKEAAAAABJRU5ErkJggg==" class="Wirisformula" alt="Error converting from MathML to accessible text." width="4" height="5" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn» «/mn»«/math»'/>

The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is (repetition is not allowed) :

Solution for the question - the sum of all the numbers that can be formed with the digits 2, 3, 4, 5taken all at a time is (repetition is not allowed) : none of these

The sum of all the numbers that can be formed with the digits 2-Turito

Detailed Solution In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2, <img src="data:image/png;base64,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" class="Wirisformula" alt="n greater or equal than 0 space." width="46" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»n«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mo».«/mo»«/math»'/> o solve this question, we should know that the total number of divisors of any number x of the form <img src="data:image/png;base64,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" class="Wirisformula" alt="a to the power of m b to the power of n c to the power of p.... space space" width="87" height="15" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»a«/mi»«mi»m«/mi»«/msup»«msup»«mi»b«/mi»«mi»n«/mi»«/msup»«msup»«mi»c«/mi»«mi»p«/mi»«/msup»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«/math»'/> where a, b, c … are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as <img src="data:image/png;base64,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" class="Wirisformula" alt="480 equals 2 to the power of 5.3.5" width="95" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»480«/mn»«mo»=«/mo»«msup»«mn»2«/mn»«mn»5«/mn»«/msup»«mo».«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«/math»'/> So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = <img src="data:image/png;base64,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" class="Wirisformula" alt="space 6 cross times 2 cross times 2 equals 24 space." width="114" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#160;«/mo»«mn»6«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»=«/mo»«mn»24«/mn»«mo»§#160;«/mo»«mo».«/mo»«/math»'/> Now, we have been asked to find the number of divisors which are of the form 4n + 2 = 2 (2n + 1), which means odd divisors cannot be a part of the solution. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1)  <img src="data:image/png;base64,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" class="Wirisformula" alt="space equals space 2 cross times 2 equals 4 space comma" width="98" height="15" style="vertical-align:-3px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»=«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mo»,«/mo»«/math»'/> according to the property. Now, we can say the total number of even divisors are = all divisors – odd divisor = 24 – 4 = 20 Now, we have been given that the divisor should be of 4n + 2, which means they should not be a multiple of 4 but multiple of 2. For that, we will subtract the multiple of 4 which are divisor of 480 from the even divisors. And, we know that,<img src="data:image/png;base64,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" class="Wirisformula" alt="480 equals 2 to the power of 5.3.5" width="95" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»480«/mn»«mo»=«/mo»«msup»«mn»2«/mn»«mn»5«/mn»«/msup»«mo».«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«/math»'/> So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) <img src="data:image/png;base64,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" class="Wirisformula" alt="equals space 4 cross times 2 cross times 2 space space equals space 16." width="139" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mo»§#160;«/mo»«mn»4«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»16«/mn»«mo».«/mo»«/math»'/> Hence, we can say that there are 16 divisors of 480 which are multiple of 4. So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4. Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, <img src="data:image/png;base64,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" class="Wirisformula" alt="space n greater or equal than 0 space." width="50" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#160;«/mo»«mi»n«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mo».«/mo»«/math»'/>

Total number of divisors of 480, that are of the form 4n + 2, n <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»'/> 0, is equal to :

Solution for the question - total number of divisors of 480, that are of the form 4n + 2, n 0, isequal to : none of these4

Total number of divisors of 480, that are of the form 4n + 2, n-Turito

If 9P5 + 5 9P4 = 10Pr , then r =

Solution for the question - if 9p5 + 59p4 = 10pr , then r = 10954

If 9p5 + 59p4 = 10pr , then r =-Turito

The number of proper divisors of <img src="data:image/png;base64,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" class="Wirisformula" alt="2 to the power of p" width="19" height="15" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mn»2«/mn»«mi»p«/mi»«/msup»«/math»'/>. <img src="data:image/png;base64,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" class="Wirisformula" alt="6 to the power of q" width="19" height="15" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mn»6«/mn»«mi»q«/mi»«/msup»«/math»'/>. 15r is-

Solution for the question - the number of proper divisors of . . 15r is- none of these(p + q) (q + r) r  2

The number of proper divisors of . . 15r is- -Turito

If<img src="data:image/png;base64,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" class="Wirisformula" alt="x squared minus 11 x plus a text  and  end text x squared minus 14 x plus 2 a" width="235" height="17" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»11«/mn»«mi»x«/mi»«mo»+«/mo»«mi»a«/mi»«mtext»§#160;and§#160;«/mtext»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»14«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mi»a«/mi»«/math»'/> have a common factor then 'a' is equal to

Solution for the question - ifhave a common factor then 'a' is equal to 122124

Ifhave a common factor then 'a' is equal to -Turito

Using conservation of linear momentum, we have Or Using conservation of energy, we have Where  compression in the spring Or

A block C of mass is moving with velocity and collides elastically with block of mass  and connected to another block of mass  through spring constant .What is  if  is compression of spring when velocity of  is same ? <img 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" alt="" />

Solution for the question - a block c of mass is moving with velocity and collides elastically withblock of mass and connected to another block of mass through

A block c of mass is moving with velocity and collides elastica-Turito

If <img src="data:image/png;base64,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" class="Wirisformula" alt="fraction numerator 1 over denominator left parenthesis 1 plus 2 x right parenthesis open parentheses 1 minus x squared close parentheses end fraction equals fraction numerator A over denominator 1 plus 2 x end fraction plus fraction numerator B over denominator 1 plus x end fraction plus fraction numerator C over denominator 1 minus x end fraction" width="321" height="40" style="vertical-align:-17px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»)«/mo»«mfenced separators=¨|¨»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mi»A«/mi»«mrow»«mn»1«/mn»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mi»B«/mi»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mi»C«/mi»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/math»'/> then ascending order of A, B, C.

Solution for the question - ifthen ascending order of a, b, c. b,a,cc,a,bb,c,aa,b,c

Ifthen ascending order of a, b, c. -Turito

The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-

Solution for the question - the number of different seven digit numbers that can be written using onlythe three digits 1, 2 and 3 with the condition that the di

The number of different seven digit numbers that can be written-Turito

The centre and radius of the circle <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHsAAAANCAYAAACJvBGjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAAmFJREFUeNrtWE1ERFEUHiNJMoyRZLRJi1ZtkiRpkyQZiRZJi0RaJYkWs2gVSau2SVpF0jJtMpK0SZIk0aLFrGIkScZQ5/DFc9x757y826T6+Bb33jfvfufnnXPvxGJ/D93EfWKeeEzsrLCeV2L8F9tXMXTCEZ/OrSfeV1BPC/HqF9tXUVwS02LukNj21+xbIi4SZ4jnxDeMo0YdcRVlpohMTAbWORs3iS/QsE2sMbwnS3wglojvCq19xC3D/A5x0JPzy2lcEnM8XiA2QCuX+AJxVrFXKPt2EYA1YrPjpe8KugJ9QVwmJojVMG4I60mUtUmUogTErhqcuIHfa8HPjxjmOdn6PQW6nEb2+bAYzyJovfBBGklfG6V9d5aHo8QKcd2xzkGdF3PN0BbEKTI5DG4cydngwVaNxjtRdnk8bXiOv+5UVPbFUTJ8gvd4FCXbtC5LNo+fxNwoKkQmxN7PIeajqGLlNEqfx9G6JKoUsQllH5/ickrHfdUBHcQTx3vbLes2bY14Pof24AJn9pFF057HBHdplHbZ7OxSxCaUfWOW5h4lBtE/bMigZ0nMocebUIMeP6G8kpgOpZOe7bZplD63xUATm1D2cR+d8mw0f7m3jvUB4oGhDHEvanX8jk/u44obwKWYS6LMVsf8w6RR+twWA01sQtm37/H6Ie+By+hDCXy13YHelA+czNPQlXW8rwPJkFLsfR24wnDynAX28gmbRulzWwy0sVHbV7DcZaNGE/7CK6G0yYztgWOKqAKmEvuEc0EJvaxVuXcbkq2IvTMe7dRolD63xUAbm++07x8/HR+6rcfwiNeXyQAAANd0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWk+cjwvbWk+PG1vPj08L21vPjxtaT5jPC9taT48bWk+bzwvbWk+PG1pPnM8L21pPjxtbz4mI3hBMDs8L21vPjxtaT4mI3gzQjg7PC9taT48bW8+LTwvbW8+PG1pPnM8L21pPjxtaT5pPC9taT48bWk+bjwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPiYjeDNCODs8L21pPjwvbWF0aD6Q+fdbAAAAAElFTkSuQmCC" class="Wirisformula" alt="r equals c o s space theta minus s i n space theta" width="123" height="13" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»r«/mi»«mo»=«/mo»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»-«/mo»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«/math»'/> are respectively

Solution for the question - the centre and radius of the circle r=cos theta-sin theta are respectively ((1)/(sqrt2),(3pi)/(4)),(1)/(sqrt2)  ((1)/(sqrt2),(7pi)/(4)),(1)/(sqrt2)

The centre and radius of the circle r=cos theta-sin theta are r-Turito

The centre of the circle <img src="data:image/png;base64,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" class="Wirisformula" alt="r squared minus 2 r left parenthesis 3 c o s space theta plus 4 s i n space theta right parenthesis minus 24" width="216" height="18" style="vertical-align:-2px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»r«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»r«/mi»«mo»(«/mo»«mn»3«/mn»«mi»c«/mi»«mi»o«/mi»«mi»s«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»+«/mo»«mn»4«/mn»«mi»s«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»§#952;«/mi»«mo»)«/mo»«mo»-«/mo»«mn»24«/mn»«/math»'/> is

Solution for the question - the centre of the circle r^(2)-2r(3cos theta+4sin theta)-24 is [9,tan^(-1)(6//5)]   [15,tan^(-1)(5//3)]   [13,tan^(-1)(5//12)]