Question
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-
- 7P2 . 25
- 7C2 . 25
- 7C2 . 52
- None of these
Hint:
Here, we will find the number of ways of the arrangement of the digit if two occur exactly twice in each seven-digit number by using the combination formula. Then by using the powers, we will find the number of ways of arranging the remaining five-digit number. We will then multiply the number of the arrangement of digits in both the cases to get the number of different seven-digit numbers.
Formula Used:
We will use the following formula:
1) Combination is given by the formula
![C presuperscript n subscript r space equals space begin inline style fraction numerator n factorial over denominator r factorial space left parenthesis n minus r right parenthesis factorial end fraction end style end subscript](data:image/png;base64,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)
2) Factorial is given by the formula
n! = n
(n-1)!
The correct answer is: 7C2 . 25
Detailed Solution
We are given the number of different seven-digit numbers that can be written using only three digits 1,2 and 3. Therefore,
![T o t a l space n u m b e r space o f space D i g i t s space equals 7](data:image/png;base64,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)
We are given that the digit two occurs exactly twice in each number.
Thus, the digit two occurs twice in the seven digit number.
Now, we will find the number of ways of arrangement of the digit two in the seven digit number by using combination.
![T o t a l space n u m b e r space o f space w a y s space t h a t space t h e space d i g i t space t w o space o c c u r s space e x a c t l y space t w i c e space i n space e a c h space n u m b e r space equals scriptbase C subscript 2 end scriptbase presuperscript 7](data:image/png;base64,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)
![N o w comma space t h e space r e m a i n i n g space f i v e space d i g i t s space c a n space b e space w r i t t e n space u sin g space t w o space d i g i t s space 1 space a n d space 3 space i n space 2 to the power of 5 space space w a y s.](data:image/png;base64,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)
We will now find the total number of seven digit number by multiplying the number of ways of arrangement in both the cases. Therefore
![T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals scriptbase C subscript 2 end scriptbase presuperscript 7 cross times space 2 to the power of 5](data:image/png;base64,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)
![N o w space b y space u sin g space t h e space f o r m u l a space scriptbase C subscript r end scriptbase presuperscript n equals fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial r factorial space end fraction space comma space w e space g e t](data:image/png;base64,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)
![rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 factorial over denominator left parenthesis 7 minus 2 right parenthesis factorial 2 factorial end fraction cross times 2 to the power of 5](data:image/png;base64,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)
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![rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 cross times 6 cross times 5 factorial over denominator 5 factorial 2 factorial end fraction cross times 2 to the power of 5](data:image/png;base64,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)
![rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 cross times 6 over denominator 2 factorial end fraction cross times 2 to the power of 5](data:image/png;base64,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)
Simplifying the expression, we get
![rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals 7 cross times 6 cross times 2 to the power of 4](data:image/png;base64,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)
Multiplying the terms, we get
![rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals 672](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWsAAAAQCAYAAADQ1F7OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAANvpXuIwAABolJREFUeNrtWw9knVcUv56IeCpERURViYiKmjAVERWlKmIiQlXETISaqZoKVRExUyqmamrMVFVViIipiDH1xMSEmpioCDUzERGqIiIivN1jv0/vTs/97v3+vDavuT8O7913z/3Ov+/ce873PaUCAiqPgqamYIaAgOq8eQ80lWPoAPPeJ/Y+wDU/dnl7Nb0GrWqqOaYxL/kqi/+qLVY/Vj3Oa5rRtKNpX9MzTT3G72UHRejWNIt1KPetaBo+igq3QbgsaNW0lJH/zyoKkmqQlxLzhqYOfG8/pola8lUW/3HerLEfYjgdriE5t+F7PRLsbx68VzT9ZHxf1DSk6YRxryxh7EhhUNPTD7zGgKbpKgqUapB3sMpsWq2+yuP+CTGcDJRMX6TkbUJCr3PMO5NlMxvD7pE3vsHaNjRqeqhpF6XGY6boJCstbhm/FbD+llFenBGuMcn4lGXOGIz9CCUclfc3PNfi49F3Wu97yLipqQ+/N2i6h2tQeXSb8Y5Dt03YpaSp2RIcs5jzNyvTTDm+RADue9jC5RNfm5oYh3yHgh9detgS1nPDnr628PGxDSfgyze4Bl3rgaCLZJuipilN2wl442I/xHDl8GOGU+8zo9p0YT+tgNHN1J2z4uSEfstvDdhdRpB467EbT7F5M9ipOeYRRA3gt82zjfM5NxDkPVjvFAxa9FiLj88gyF7A8TXYSP5CwFJpdBXjp3GdJoP3FQIy0u2ecFKph/36jRPBqiAXtSu+09Ti4S8fn8yxJDLkEVtUFtZafnfp0Sq00shHyyls4eNjCQUkG9MuUTIdcMRCDfx9C5+JbmLjGvCIowHPey3E8P9RVv59ZRNrlk3FBdpMJjzndrnaW2VPuqPyezDwKkbxKQStCXLGumA8fmIeghNNPGYnrQjrCNo4rKNPxUGnhpMea/HxdZz8Gti8DSQuabzZ4G0XgnpHsN9tYQOrZXIMJvBXEp+c9lyTgvKS45ouPfjDqt/Zmr628PGxhAnLaU6KBz42gXtK4m128K5ZqsUQw5UDbTptOJTsoWqnDWvY0dZYVn4P2eswN/PB+CESdh59shqU0raTyrZQXtehzDTn7Qn8JexOJhaFwLbx8zm7Fvn3PNYqWOYVLfOKMfy2a9SxQC/ATo1sbMdIaj665+kTG+iByx+WCstHjyg5f4LPvfB1Ulv4+NiGf9TbB0RxdpDGNoXNQJKn4BlvIYYrizLitQ/2KyCx0qbxlYWH4vGiZ+X6s6bLeZQFRHdzOll3IalK+FTJT1Y7GU8nuzFtJy1bsHbGyOCa0yXIUvKUuZTgOuZ4nCzPGY/ku6WEuqf1SSlhLDRj7RJLej56RFXTFXxeYRt1Flt0eejSYYlBaU0+5mvTJLEfYriybZAdS+XRgepBOowseMjTgkTdmjWxVqJnPYT+mYR+oY1B+JqVjEO4UTm22fdu7IZJZHDN4eO2eV94zvMZt825yfphNvsl1T2tTx6liIc69Cc/T6iHwolmCj3VXz3lVhl8YoJK8CeevHwsC68t9kMMVxYLMYfVA6GiWFPuh4pn0Toq5iHgmFDmZQU9/Bu1/NYr7Eak+EsoFuG+pusC/wYr1ykIZi0yXEspJx8nWfiT9fMoc0c9rukzTp9H2O+1SHLN7JSynMH+WXziY9O4NttwQj0USlLqIa4KN0YWW/jYaMCSVJ4IduDrUUKatvCOesTb9Yz32nGL4TxAB4OrloS7LMSGq/ppQvwc6T+NzSn5gV/US6OE+xm+n8L8cTbvAW5wjm/Vf6/YUDKhPvVTSykSJ4NrDh+fxEmngD7bJOTj83zXk8bn0Bu7xOwyIvC9xKm3gF4YPai5kFD3ND5Juq6ZFEjmkwn1UDiRRP8iUznawkeXItbvMFob87BVn2M9eqi2ZfB24FBBvD0OXlvshxiuLGqRgEeNBNuJzYY/KJ+26GVinh12jiReq/iXwy/AWQcoJSSl2xHYZfXuq1/j2LEmcJ0t9e4rSS4Z4ubwcQqmXyDvopHU+Dzf9aTxLZzGKNjp1a4VZX91qwX90EPYcTSF7ml8kmTdN/DdIXqPZ1PoYa51Lmdb+OrShMRBGwY97Lxo4ZXG+tFijN4quIw2XsHBGxf7IYYri0a0LXYh26JwgIh0bXCslbRnfizQqALeh403gxky4ZwK//4MCAioIO7jtH03mMIbVEn8oN72aqkVtFQNZXFAQED1gnq1g8EMiRD9RZ3KaXrddEHZWzkBxxT/AokNx7G94LpWAAACAHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4mI3gyMUQyOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPlQ8L21pPjxtaT5vPC9taT48bWk+dDwvbWk+PG1pPmE8L21pPjxtaT5sPC9taT48bW8+JiN4QTA7PC9tbz48bWk+bjwvbWk+PG1pPnU8L21pPjxtaT5tPC9taT48bWk+YjwvbWk+PG1pPmU8L21pPjxtaT5yPC9taT48bW8+JiN4QTA7PC9tbz48bWk+bzwvbWk+PG1pPmY8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5zPC9taT48bWk+ZTwvbWk+PG1pPnY8L21pPjxtaT5lPC9taT48bWk+bjwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmQ8L21pPjxtaT5pPC9taT48bWk+ZzwvbWk+PG1pPmk8L21pPjxtaT50PC9taT48bW8+JiN4QTA7PC9tbz48bWk+bjwvbWk+PG1pPnU8L21pPjxtaT5tPC9taT48bWk+YjwvbWk+PG1pPmU8L21pPjxtaT5yPC9taT48bW8+JiN4QTA7PC9tbz48bW8+PTwvbW8+PG1uPjY3MjwvbW4+PC9tYXRoPk4jREIAAAAASUVORK5CYII=)
Therefore, the number of different seven-digit numbers that can be written using only three digits 1,2 and 3 is 672.
We know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.
Related Questions to study
The centre and radius of the circle
are respectively
The centre and radius of the circle
are respectively
The centre of the circle
is
The centre of the circle
is
The equation of the circle with centre at
, which passes through the point
is
The equation of the circle with centre at
, which passes through the point
is
The foot of the perpendicular from
on the line
is
The foot of the perpendicular from
on the line
is
The foot of the perpendicular from the pole on the line
is
The foot of the perpendicular from the pole on the line
is
The equation of the line parallel to
and passing through
is
The equation of the line parallel to
and passing through
is
The line passing through the points
, (3,0) is
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution is .
The line passing through the points
, (3,0) is
So here we used the concept of the equation of the line passing through two points. Here we also used the trigonometric terms to find the answer using the formulas. So the final solution is .
Statement-I : If
then A=![7 over 4](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJ9JREFUeNpjYMAO/hPAFINQIJ5NqSHiQHwYiDkoNWgTEBtQakgGENdSaog8EJ8EYhZKDToIxI7UiKVtlBrCBMQ3qBHAAVBvUQyWA3EiNQx6CcSCDKNgcIP/ZOJRMFiBDzViiAeIr1HDoJlAnEypQdZAvBcpwZIF2ID4ErQWocigDiDOQctCJAM9ID6KJS+SDEAVogo1DKJpjqdakUEfgwCGRTp0g/2KWQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjc8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7kU9SNAAAAAElFTkSuQmCC)
Statement-II : If
then ![p equals 2 comma q equals 3](data:image/png;base64,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)
Which of the above statements is true
Statement-I : If
then A=![7 over 4](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJ9JREFUeNpjYMAO/hPAFINQIJ5NqSHiQHwYiDkoNWgTEBtQakgGENdSaog8EJ8EYhZKDToIxI7UiKVtlBrCBMQ3qBHAAVBvUQyWA3EiNQx6CcSCDKNgcIP/ZOJRMFiBDzViiAeIr1HDoJlAnEypQdZAvBcpwZIF2ID4ErQWocigDiDOQctCJAM9ID6KJS+SDEAVogo1DKJpjqdakUEfgwCGRTp0g/2KWQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjc8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7kU9SNAAAAAElFTkSuQmCC)
Statement-II : If
then ![p equals 2 comma q equals 3](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFEAAAAPCAYAAACcCyOxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAAepJREFUeNrVl7FLw0AUxkORIlIK4lCkg1AcxEEKDiIiIohDcQiCgziULs7i4iAi/gMixc3BwSEgIg4iLiIOIm4iIlIQBxG3Ih1KEUG/B9+Qhou9SC6JH/ygueaS9+69d+9iWfFoAhyBBvgEd2DJSqamwRloghaogR3QF7dhV2ARZHg9DK45ljRdgnmQdo0VwXkSIz4A7q3/o4ZqcBOsgy3wztSVKPRHaFgrwL2SxVXwwXmyPeyCtQjsLIAn1R+H4JlG9IIU2AbOLw/71kBX4yxpHaUY4Ap/Z5kE8j7boK05vlP2xZLqhhr3JreyfmkbsrrBLRuOjjZ8Mk58yBuwz7vYK36Rbfo4Z3oRJetPwGyAOa+uptTJh7BttRnwKe+fYywPVYldGCznAhdwMIAjRXZ3XR9MbD0ZLmSb5Gixr7h5laVjQkNgD/QEnCfHjQPFuJ8PkTXBKjdMt9I8cpjozjk2sq4/zLU51ytZ2OWIFnCE+2+bjjk4w+s8xyqGjDhlJnaSo+i4krmPLGvRKJ/35tcxQ/gwWGDAU/yCeQFl7411GioL+cXPMNtgJHX3Jcfn2JJjkKWkbuhYnY0wbE0ySC3X2XnOUpRtPcFfBo5GhqV56I5NJUY1iZKSfdDYO2P3oczGkkTlWbqJ9uEHLUyOob+pMncAAACFdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPnA8L21pPjxtbz49PC9tbz48bW4+MjwvbW4+PG1vPiw8L21vPjxtaT5xPC9taT48bW8+PTwvbW8+PG1uPjM8L21uPjwvbWF0aD7FLdr5AAAAAElFTkSuQmCC)
Which of the above statements is true
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If b > a , then the equation, (x - a) (x - b) - 1 = 0, has:
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, one of the roots will be in the interval of (−α,a) and the other root will be in the interval (b,α).
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
If
be that roots
where
, such that
and
then the number of integral solutions of λ is
Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.
If α,β then the equation
with roots
will be
Here we used the concept of quadratic equations and solved the problem. We found the sum and product of the roots first and then proceeded for the final answer. Therefore, will be the equation for the roots
.