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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 51-100 of 303

  1. If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word …
  2. From a group of $7$ batsmen and $6$ bowlers, $10$ players are to be chosen for a team, which should include atleast $4$ batsmen and atleast $4$ bowlers. One batsmen and …
  3. The number of ways to distribute $30$ identical candies among four children $C_1,C_2,C_3$ and $C_4$ so that $C_2$ receives atleast $4$ and atmost $7$ candies, $C_3$ …
  4. The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is
  5. If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8$, then the number of all numbers greater than $10{,}000$ is equal to …
  6. If the number of five digit numbers with distinct digits and $2$ at the $10^{\text{th}}$ place is $336k$, then $k$ is equal to :
  7. Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that …
  8. Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A\times B$ each having at least $3$ and at most $6$ elements …
  9. Team 'A' consists of $7$ boys and $n$ girls and Team 'B' has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy …
  10. The sum of all the elements in the set $\{n\in\{1,2,\ldots,100\}\mid\text{H.C.F. of }n\text{ and }2040\text{ is }1\}$ is equal to $\_\_\_\_\_\_$ .
  11. The number of permutations, of the digits $1,2,3,\ldots,7$ without repetition, which neither contain the string $153$ nor the string $2467$, is $\_\_\_\_\_\_$ .
  12. The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to $\_\_\_\_\_\_$ .
  13. Let $S=\{1,2,3,4,5,6,9\}$. Then the number of elements in the set $T=\{A\subseteq S:A\neq\phi$ and the sum of all the elements of $A$ is not a multiple of $3\}$ is …
  14. In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections: $A,B$ and $C$. A student is …
  15. In an examination, there are $5$ multiple choice questions with $3$ choices, out of which exactly one is correct. There are $3$ marks for each correct answer, $-2$ marks …
  16. Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{A,B,C,D,E\}$ or a number from …
  17. The number of ways, $16$ identical cubes, of which $11$ are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least $2$ …
  18. If ${}^{1}P_1+2\cdot{}^{2}P_2+3\cdot{}^{3}P_3+\ldots+15\cdot{}^{15}P_{15}={}^{q}P_r-s$, $0\le s\le1$, then ${}^{q+s}C_{r-s}$ is equal to $\_\_\_\_\_\_$ .
  19. The sum of all the elements of the set $\{\alpha\in\{1,2,\ldots,100\}:\text{HCF}(\alpha,24)=1\}$ is $\_\_\_\_\_\_$ .
  20. A person forgets his $4$-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is $7$ and the sum of the first two …
  21. The number of $3$-digit odd numbers, whose sum of digits is a multiple of $7$, is $\_\_\_\_\_\_$ .
  22. An urn contains $5$ red marbles, $4$ black marbles and $3$ white marbles. Then the number of ways in which $4$ marbles can be drawn so that at the most three of them are …
  23. Let $A=\{1,2,3,4,5\}$ and $B=\{1,2,3,4,5,6\}$. Then the number of functions $f:A\to B$ satisfying $f(1)+f(2)=f(4)-1$ is equal to $\_\_\_\_\_\_$ .
  24. The number of $4$-digit numbers which are neither multiple of $7$ nor multiple of $3$ is $\_\_\_\_\_\_$ .
  25. The number of one-one functions $f:\{a,b,c,d\}\to\{0,1,2,\ldots,10\}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is $\_\_\_\_\_\_$ .
  26. All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the …
  27. The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to $\_\_\_\_\_\_$ .
  28. Let $n$ be a non-negative integer. Then the number of divisors of the form ``$4n+1$'' of the number $(10)^{10}\cdot(11)^{11}\cdot(13)^{13}$ is equal to ________.
  29. The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $\_\_\_\_\_\_$ .
  30. The number of $4$-letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word UNIVERSE …
  31. The number of $3$-digit numbers, formed using the digits $2,3,4,5$ and $7$, when the repetition of digits is not allowed, and which are not divisible by $3$, is equal to …
  32. Let $m$ and $n$, $(m<n)$, be two $2$-digit numbers. Then the total numbers of pairs $(m,n)$, such that $\gcd(m,n)=6$, is $\_\_\_\_\_\_$ .
  33. There are ten boys $B_1,B_2,\ldots,B_{10}$ and five girls $G_1,G_2,\ldots,G_5$ in a class. Then the number of ways of forming a group consisting of three boys and three …
  34. The number of elements in the set $S=\{(x,y,z):x,y,z\in\mathbf{Z},\ x+2y+3z=42,\ x,y,z\ge0\}$ equals $\_\_\_\_\_\_$ .
  35. A class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$, then $b+3g$ is equal to ________.
  36. The total number of three-digit numbers, with one digit repeated exactly two times, is $\_\_\_\_\_\_$ .
  37. Numbers are to be formed between $1000$ and $3000$, which are divisible by $4$, using the digits $1,2,3,4,5$ and $6$ without repetition of digits. Then the total number …
  38. The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is $\_\_\_\_\_\_$ .
  39. Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1,2,3,4,5$ with repetition, is $\_\_\_\_\_\_$ .
  40. The number of $5$-digit natural numbers, such that the product of their digits is $36$, is ________.
  41. The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is $\_\_\_\_\_\_$ .
  42. The number of $6$-digit numbers made by using the digits $1,2,3,4,5,6,7$, without repetition and which are multiple of $15$ is $\_\_\_\_\_\_$ .
  43. The lines $L_1,L_2,\ldots,L_{20}$ are distinct. For $n=1,2,3,\ldots,10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a …
  44. The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is $\_\_\_\_\_\_$ .
  45. The number of matrices of order $3\times3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is ________.
  46. Let $b_1b_2b_3b_4$ be a $4$-element permutation with $b_i\in\{1,2,3,\ldots,100\}$ for $1\le i\le4$ and $b_i\neq b_j$ for $i\neq j$, such that either $b_1,b_2,b_3$ are …
  47. The total number of $3$-digit numbers, whose greatest common divisor with $36$ is $2$, is $\_\_\_\_\_\_$ .
  48. In an examination, $5$ students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is …
  49. Let $A$ be a $3\times3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is …
  50. The letters of the word `MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word …

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