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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 151-200 of 303

  1. The total number of $4$-digit numbers whose greatest common divisor with $54$ is $2$, is
  2. Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is NOT in $S$. Then the number …
  3. Six X's have to be placed in the squares of the figure below in such a way that each row contains at least one X. In how many different ways can this be done? [Figure: a …
  4. Five balls of different colours are to be placed in three boxes of different size. Each box can hold all five. In how many different ways can we place the balls so that …
  5. Let $S=\{1,2,3,5,7,10,11\}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is
  6. Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets $$A=\{1,2,3,4,5\}\text{ and }B=\{1,2,3,4,5,6,7\}.$$ Let $S$ be the set of all functions …
  7. In a certain test, $a_i$ students gave wrong answers to at least $i$ questions, where $i=1,2,\ldots,k$. No student gave more than $k$ wrong answers. The total number of …
  8. All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two $R$ appearing together. The arrangements are listed …
  9. If all the six digit numbers $x_1x_2x_3x_4x_5x_6$ with $0<x_1<x_2<x_3<x_4<x_5<x_6$ are arranged in the increasing order, then the sum of the digits in the …
  10. Let $x$ and $y$ be distinct integers where $1\le x\le25$ and $1\le y\le25$. Then, the number of ways of choosing $x$ and $y$, such that $x+y$ is divisible by $5$, is
  11. A group of $9$ students, $s_1,s_2,\ldots,s_9$, is to be divided to form three teams $X,Y$, and $Z$ of sizes $2,3$, and $4$, respectively. Suppose that $s_1$ cannot be …
  12. The number of ways to distribute $10$ identical red pens and $14$ identical blue pens among four persons such that each person gets $6$ pens, is ______________.
  13. There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicketkeepers. The number of ways, a team of $11$ players be selected …
  14. If ${}^{n}C_{r-1}=36$, ${}^{n}C_{r}=84$ and ${}^{n}C_{r+1}=126$, then $r$ is
  15. Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number $77777$ has …
  16. Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked $1$ to $4$ …
  17. If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
  18. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
  19. $8$-digit numbers are formed using the digits $1,1,2,2,2,3,4,4$. The number of such numbers in which the odd digits do not occupy odd places, is :
  20. How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order
  21. The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by
  22. Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that …
  23. The number of ways of distributing $8$ identical balls in $3$ distinct boxes so that none of the boxes is empty is
  24. At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are to be selected, if a …
  25. Total number of four digit odd numbers that can be formed using $0,1,2,3,5,7$ (using repetition allowed) are :
  26. If ${}^{n}C_r$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2\times{}^{n}C_{r}$ equals
  27. The sum of the digits in the unit's place of all the $4$-digit numbers formed by using the numbers $3,4,5$ and $6$, without repetition is :
  28. If $\frac{{}^{n+2}C_{6}}{{}^{n-2}P_{2}}=11$, then $n$ satisfies the equation :
  29. The number of ways in which $5$ boys and $3$ girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other …
  30. The range of the function $f(x)={}^{7-x}P_{x-3}$ is
  31. In a shop there are five types of ice-cream available. A child buys six ice-cream. Statement - 1: The number of different ways the child can buy the six ice-cream is …
  32. The set $S=\{1,2,3,\ldots,12\}$ is to be partitioned into three sets $A,B,C$ of equal size. Thus $A\cup B\cup C=S$, $A\cap B=B\cap C=A\cap C=\phi$. The number of ways to …
  33. Five digit numbers divisible by $3$ are formed using $0,1,2,3,4$ and $5$ without repetition. Total number of such numbers are :
  34. Numbers greater than $1000$ but less than $4000$ are formed using the digits $0,1,2,3,4$ (repetition allowed). Their number is :
  35. A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no …
  36. A student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first five questions. The number of choices available …
  37. 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 …
  38. An eight digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is :
  39. The number of four-digit numbers strictly greater than $4321$ that can be formed using the digits $0,1,2,3,4,5$ (repetition of digits is allowed) is :
  40. A committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the …
  41. The number of functions $f$ from $\{1,2,3,\ldots,20\}$ onto $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$ is :
  42. Total number of $6$-digit numbers in which only and all the five digits $1,3,5,7$ and $9$ appears, is
  43. If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN …
  44. The number of ordered pairs $(r,k)$ for which $6\cdot{}^{35}C_r=\left(k^2-3\right)\cdot{}^{36}C_{r+1}$, where $k$ is an integer is :
  45. All possible numbers are formed using the digits $1,1,2,2,2,2,3,4,4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is
  46. The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :
  47. The number of $9$ digit numbers, that can be formed using all the digits of the number $123412341$ so that the even digits occupy only even places, is ______
  48. The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ______.
  49. A committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$ young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ …
  50. $5$-digit numbers are to be formed using $2,3,5,7,9$ without repeating the digits. If $p$ be the number of such numbers that exceed $20000$ and $q$ be the number of …

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