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Algebra

Permutation and Combination: JEE Main & Advanced PYQs

303 previous year questions on Permutation and Combination from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 303

  1. A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the …
  2. If $n\ge2$ is a positive integer, then the sum of the series ${}^{n+1}C_{2}+2\left({}^{2}C_{2}+{}^{3}C_{2}+{}^{4}C_{2}+\ldots+{}^{n}C_{2}\right)$ is :
  3. Let $A=\{(a,b,c):a,b,c$ are non-negative integers and $a+b+2c=22\}$. Then $n(A)$ is equal to:
  4. A building has ground floor and $10$ more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. The number of ways …
  5. A box contains $5$ blue, $6$ yellow and $4$ red balls. The number of ways, of drawing $8$ balls containing at least two balls of each colour, is :
  6. The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1,2$ and $3$ only is :
  7. The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :
  8. If for $3\le r\le 30$, $\left({}^{30}C_{30-r}\right)+3\left({}^{30}C_{31-r}\right)+3\left({}^{30}C_{32-r}\right)+\left({}^{30}C_{33-r}\right)={}^{m}C_{r}$, then $m$ …
  9. Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-side regular polygon. If $p_{n+1}-p_n=66$, then the sum of all distinct prime …
  10. A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is :
  11. Consider a rectangle $ABCD$ having $5,7,6,9$ points in the interior of the line segments $AB$, $CD$, $BC$, $DA$ respectively. Let $\alpha$ be the number of triangles …
  12. A natural number has prime factorization given by $n=2^{x}3^{y}5^{z}$, where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}$, $y>z$. Then the number of …
  13. All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number …
  14. If ${}^{n}P_r={}^{n}P_{r+1}$ and ${}^{n}C_r={}^{n}C_{r-1}$, then the value of $r$ is equal to :
  15. Let $M$ be any $3\times3$ matrix with entries from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $M^{T}M$ is seven …
  16. Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of …
  17. If the sides $AB$, $BC$ and $CA$ of a triangle $ABC$ have $3,5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using …
  18. The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :
  19. ${}^{n-1}C_r=\left(k^{2}-8\right){}^{n}C_{r+1}$ if and only if :
  20. The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5$, if the repetition of digits is not allowed and numbers are …
  21. If the number of permutations of the word "STATISTICS" in which all '$S$' do not come together, is $(56)^{2}k$, then '$k$' is equal to $\_\_\_\_\_\_$ .
  22. Let $n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. …
  23. There are $5$ students in class $10$, $6$ students in class $11$ and $8$ students in class $12$. If the number of ways, in which $10$ students can be selected from them …
  24. If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then …
  25. The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$, is $\_\_\_\_\_\_$ .
  26. The largest $n\in\mathbf{N}$ such that $3^{n}$ divides $50!$ is :
  27. Let $0\le r\le n$. If ${}^{n+1}C_{r+1}:{}^{n}C_r:{}^{n-1}C_{r-1}=55:35:21$, then $2n+5r$ is equal to :
  28. The total number of $5$-digit numbers, formed by using the digits $1,2,3,5,6,7$ without repetition, which are multiple of $6$, is
  29. The value of $(2\cdot{}^1P_0-3\cdot{}^2P_1+4\cdot{}^3P_2-\ldots$ up to $51^{\text{th}}$ term$)+(1!-2!+3!-\ldots$ up to $51^{\text{th}}$ term$)$ is equal to :
  30. The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is $\_\_\_\_\_\_$ .
  31. The number of triplets $(x,y,z)$, where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is
  32. The students $S_1,S_2,\ldots,S_{10}$ are to be divided into $3$ groups $A$, $B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ …
  33. The sum of all values of $n$, for which ${}^{n-1}C_4-{}^{n-1}C_3-\frac54\,{}^{n-2}P_2<0$, is
  34. If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text{th}}$ position in …
  35. The number of bijective functions $f:\{1,3,5,7,\ldots,99\}\to\{2,4,6,8,\ldots,100\}$, such that $f(3)\ge f(9)\ge f(15)\ge f(21)\ge\ldots\ge f(99)$, is
  36. The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is
  37. All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the …
  38. Line $L_1$ of slope $2$ and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant, $P_1,P_2,\ldots,P_{12}$ are $12$ points on line $L_1$ …
  39. Let $A=\{n\in[100,700]\cap\mathbb{N}:n$ is neither a multiple of $3$ nor a multiple of $4\}$. Then the number of elements in $A$ is
  40. If six letter words (with or without meaning) are formed using $A,B,C,D,E$ and $F$, then the number of words, in which exactly one of these alphabets is repeated three …
  41. If ${}^{2n}C_3:{}^{n}C_3=10:1$, then the ratio $\left(n^{2}+3n\right):\left(n^{2}-3n+4\right)$ is
  42. The number of five digit numbers, greater than $40000$ and divisible by $5$, which can be formed using the digits $0,1,3,5,7$ and $9$ without repetition, is equal to
  43. Each of the numbers $1,0,-1$ is written on two different opposite faces of a six faces fair die. This die is thrown three times. The number of ways in which the sum of …
  44. The total number of three-digit numbers, divisible by $3$, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is
  45. The number of sequences of ten terms, whose terms are either $0$ or $1$ or $2$, that contain exactly five $1$s and exactly three $2$s, is equal to :
  46. The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples, is
  47. If for some $m,n$; ${}^{6}C_m+2\left({}^{6}C_{m+1}\right)+{}^{6}C_{m+2}>{}^{8}C_3$ and ${}^{n-1}P_3:{}^{n}P_4=1:8$, then ${}^{n}P_{m+1}+{}^{n+1}C_m$ is equal to
  48. If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $(6!)k$, then …
  49. The sum of all the $4$-digit distinct numbers that can be formed with the digits $1,2,2$ and $3$ is :
  50. The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

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