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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 101-150 of 303

  1. The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to $\_\_\_\_\_\_$ .
  2. The largest natural number $n$ such that $3^{n}$ divides $66!$ is $\_\_\_\_\_\_$ .
  3. The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by …
  4. Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is $840$, then the total numbers …
  5. If the number of seven-digit numbers, such that the sum of their digits is even, is $m\cdot n\cdot10^{n}$; $m,n\in\{1,2,3,\ldots,9\}$, then $m+n$ is equal to …
  6. The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is …
  7. The number of integers, between $100$ and $1000$ having the sum of their digits equals to $14$, is $\_\_\_\_\_\_$ .
  8. A number is called a palindrome if it reads the same backward as well as forward. For example $285582$ is a six digit palindrome. The number of six digit palindromes …
  9. For $n\ge2$, let $S_n$ denote the set of all subsets of $\{1,2,\ldots,n\}$ with no two consecutive numbers. For example $\{1,3,5\}\in S_6$ but $\{1,2,4\}\notin S_6$. …
  10. The number of words, with or without meaning, that can be formed by taking $4$ letters at a time from the letters of the word `SYLLABUS' such that two letters are …
  11. The number of words (with or without meaning) that can be formed from all the letters of the word ``LETTER'' in which vowels never come together is ________.
  12. The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word `EXAMINATION' is ________.
  13. A test consists of $6$ multiple choice questions, each having $4$ alternative answers of which only one is correct. The number of ways, in which a candidate answers all …
  14. The total number of $3$-digit numbers, whose sum of digits is $10$, is ________.
  15. If the letters of the word `MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word …
  16. From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is …
  17. The number of four letter words that can be formed using the letters of the word BARRACK is
  18. Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to
  19. $n$-digit numbers are formed using only three digits $2$, $5$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed, is
  20. Let $A=\{x_1,x_2,\ldots,x_7\}$ and $B=\{y_1,y_2,y_3\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions …
  21. Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from …
  22. The number of ways of selecting two numbers $a$ and $b$, $a\in\{2,4,6,\ldots,100\}$ and $b\in\{1,3,5,\ldots,99\}$ such that $2$ is the remainder when $a+b$ is divided by …
  23. There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the …
  24. The number of 6 digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by $11$ and no digit is repeated is :
  25. Consider a class of $5$ girls and $7$ boys. The number of different teams consisting of $2$ girls and $3$ boys that can be formed from this class, if there are two …
  26. If in a regular polygon the number of diagonals is $54$, then the number of sides of this polygon is :
  27. Let $P$ be the set of seven digit numbers with sum of their digits equal to $11$. If the numbers in $P$ are formed by using the digits $1,2$ and $3$ only, then the …
  28. A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there …
  29. Let $S=\{1,2,3,\ldots,100\}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :
  30. Let $\alpha=\frac{(4!)!}{(4!)^{3!}}$ and $\beta=\frac{(5!)!}{(5!)^{4!}}$. Then :
  31. If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{th}$ position in this …
  32. If ${}^{n}C_4$, ${}^{n}C_5$ and ${}^{n}C_6$ are in A.P., then $n$ can be
  33. The number of natural numbers less than $7000$ which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to :
  34. Consider three boxes, each containing $10$ balls labelled $1,2,\ldots,10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$, the label of the …
  35. The number of ways of selecting $15$ teams from $15$ men and $15$ women, such that each team consists of a man and a woman is
  36. The number of ways of choosing $10$ objects out of $31$ objects of which $10$ are identical and the remaining $21$ are distinct, is:
  37. Let $P_1,P_2,\ldots,P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_i,P_j,P_k$ such that $i+j+k\neq15$, is :
  38. The number of 4-digit integers in the closed interval $[2022,\,4482]$ formed by using the digits $0,2,3,4,6,7$ is __________.
  39. The number of $3$ digit numbers, that are divisible by either $3$ or $4$ but not divisible by $48$, is
  40. If all the words (with or without meaning) having five letters, formed using the letters of the word $SMALL$ and arranged as in a dictionary, then the position of the …
  41. The number of integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is
  42. The number of numbers, strictly between $5000$ and $10000$ can be formed using the digits $1,3,5,7,9$ without repetition, is :
  43. Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top …
  44. If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is R and the fourth letter …
  45. Five persons $A, B, C, D$ and $E$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number …
  46. Let $A=\{1,2,3,4,5,6\}$. The number of one-one functions $f:A\to A$ such that $f(1)\ge3$, $f(3)\le4$ and $f(2)+f(3)=5$, is ________.
  47. Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least …
  48. The value of the expression ${}^{47}C_4+\displaystyle\sum_{j=1}^{5}{}^{52-j}C_3$ is equal to
  49. Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter …
  50. Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of …

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