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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 251-300 of 303

  1. The number of ways, of forming a queue of $4$ boys and $3$ girls such that all the girls are not together, is :
  2. The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3,\ldots,9\}$ such that $f(i)\ne i$ for $1\le i\le 6$, is equal to :
  3. The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word …
  4. Let $S=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated …
  5. The largest value of $n$, for which $40^n$ divides $60!$, is :
  6. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear …
  7. Let $n\geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent …
  8. Six cards and six envelopes are numbered $1,2,3,4,5,6$ and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in …
  9. Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$. Then the number of such distinct arrangements $(n_1,n_2,n_3,n_4,n_5)$ is
  10. 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
  11. Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to
  12. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
  13. Let $n$ be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number …
  14. A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for …
  15. Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number …
  16. Let $S=\{1,2,3,\dots,9\}$. For $k=1,2,\dots,5$, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then …
  17. The number of 5 digit numbers which are divisible by 4, with digits from the set $\{1,2,3,4,5\}$ and the repetition of digits is allowed, is _____ .
  18. In a high school, a committee has to be formed from a group of 6 boys $M_1,M_2,M_3,M_4,M_5,M_6$ and 5 girls $G_1,G_2,G_3,G_4,G_5$. (i) Let $\alpha_1$ be the total number …
  19. Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the …
  20. An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive …
  21. In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at …
  22. The number of different 5 digit numbers greater than $50000$ that can be formed using the digits $0, 1, 2, 3, 4, 5, 6, 7$, such that the sum of their first and last …
  23. The number of ways, $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together, is ______.
  24. If $n(X)={}^{m}C_6$, then the value of $m$ is __________.
  25. If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is _____________.
  26. The value of $b_6$ is
  27. Which of the following is correct?
  28. State TRUE or FALSE: If $n_1,n_2,\ldots,n_p$ are $p$ positive integers whose sum is an even number, then the number of odd integers among them is odd.
  29. Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II …
  30. $m$ men and $n$ women are to be seated in a row so that no two women sit together. If $m>n$, then show that the number of ways in which they can be seated is …
  31. Using induction or otherwise, prove that for any non-negative integers $m,n,r$ and $k$ …
  32. Let $p$ be a prime and $m$ a positive integer. By mathematical induction on $m$, or otherwise, prove that whenever $r$ is an integer such that $p$ does not divide $r$ …
  33. Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from $10$ white, $9$ green and $7$ …
  34. Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X$, $Z\subseteq X$ and $Y\cap Z$ is empty, is :
  35. $mn$ squares of equal size are arranged to form a rectangle of dimension $m$ by $n$, where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if …
  36. Prove by permutation or otherwise that $\dfrac{(n^{2})!}{(n!)^{n}}$ is an integer $(n\in I^{+})$.
  37. The unit digit in the sum of $1!+2!+3!+\ldots+2021!$ is $\_\_\_\_\_\_\_\_$ .
  38. The total number of $3\times3$ matrices $A$ having entries from the set $\{0,1,2,3\}$ such that the sum of all the diagonal entries of $AA^{T}$ is $9$, is equal to …
  39. There are $5$ points $P_1,P_2,P_3,P_4,P_5$ on the side $AB$, excluding $A$ and $B$, of a triangle $ABC$. Similarly there are $6$ points $P_6,P_7,\ldots,P_{11}$ on the …
  40. There are $4$ men and $5$ women in Group A, and $5$ men and $4$ women in Group B. If $4$ persons are selected from each group, then the number of ways of selecting $4$ …
  41. The number of ways of getting a sum $16$ on throwing a dice four times is $\_\_\_\_\_\_\_\_$ .
  42. $60$ words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text{th}}$ word is …
  43. Let the set $S=\{2,4,8,16,\ldots,512\}$ be partitioned into $3$ sets $A,B,C$ with equal number of elements such that $A\cup B\cup C=S$ and …
  44. There are $3$ sections in a question paper and each section contains $5$ questions. A candidate has to answer a total of $5$ questions, choosing at least one question …
  45. Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are …
  46. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :
  47. In a group of $3$ girls and $4$ boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls …
  48. The number of words, which can be formed using all the letters of the word ``DAUGHTER'', so that all the vowels never come together, is :
  49. Group A consists of $7$ boys and $3$ girls, while group B consists of $6$ boys and $5$ girls. The number of ways, $4$ boys and $4$ girls can be invited for a picnic if …
  50. Number of functions $f:\{1,2,\ldots,100\}\to\{0,1\}$, that assign $1$ to exactly one of the positive integers less than or equal to $98$, is equal to $\_\_\_\_\_\_\_\_$ .

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