Permutations and Combinations jee mains questions | JEE Main Maths Permutations and Combinations Previous Year Question

Q 41 :

The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ________.

(2454)

The word EXAMINATION has $2 N, 2 A, 2 I, E, X, M, T, O$

Case I: If all letters are different, then number of words

$= {}^{8}P_{4} = \frac{8!}{4!} = 8 \times 7 \times 6 \times 5 = 1680$

Case II: If 2 letters are same and 2 are different, then number of words

$= {}^{3}C_{1} \cdot {}^{7}C_{2} \cdot \frac{4!}{2!} = 3 \times 21 \times 12 = 756$

Case III: If 2 letters are same of one kind and 2 letters are same of different kind, then number of words

$= {}^{3}C_{2} \cdot \frac{4!}{2! 2!} = 3 \times 6 = 18$

$∴$ Total number of words $= 1680 + 756 + 18 = 2454$

Q 42 :

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 distinct and two letters are alike, is ________.

(240)

$The word SYLLABUS consists of$

$2 S, 2 L, 1 Y, 1 A, 1 B, 1 U$

$Required number of ways = {}^{2}C_{1} \times {}^{5}C_{2} \times \frac{4!}{2!}$

$= 2 \times 10 \times 12 = 240$

Q 43 :

The value of ${}^{16}C_{9} + {}^{16}C_{10} - {}^{16}C_{6} - {}^{16}C_{7}$ is

1
${}^{17}C_{10}$
${}^{17}C_{3}$
$0$

1

2

3

4

(4)

We have, ${}^{16}C_{9} + {}^{16}C_{10} - {}^{16}C_{6} - {}^{16}C_{7}$

$= {}^{17}C_{10} - {}^{17}C_{7}$ $(∵ {}^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r})$

$= {}^{17}C_{10} - {}^{17}C_{10}$ $(∵ {}^{n}C_{r} = {}^{n}C_{n - r})$

$= 0$

Q 44 :

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicket keepers. The number of ways a team of 11 players can be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicket keeper is ________.

(777)

Total number of players = 15

Number of bowlers = 6

Number of batsmen = 7

Number of wicket keepers = 2

$∴$ Total number of ways for which at least 4 bowlers, 5 batsmen and 1 wicket keeper is to be selected

$= {}^{6}C_{4} {}^{7}C_{5} {}^{2}C_{1} + {}^{6}C_{4} {}^{7}C_{6} {}^{2}C_{1} + {}^{6}C_{5} {}^{7}C_{5} {}^{2}C_{1}$

$= 315 + 210 + 252 = 777$

Q 45 :

The value of $\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to

${}^{51}{C_{3}} - {}^{45}{C_{3}}$
${}^{52}C_{4} - {}^{45}C_{4}$
${}^{52}C_{3} - {}^{45}C_{3}$
${}^{51}C_{4} - {}^{45}C_{4}$

1

2

3

4

(2)

$\sum_{k = 0}^{6} {}^{51 - k}C_{3} = {}^{51}C_{3} + {}^{50}C_{3} + {}^{49}C_{3} + {}^{48}C_{3} + {}^{47}C_{3} + {}^{46}C_{3} + {}^{45}C_{3}$

As we know, ${}^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r} \Rightarrow {}^{n}C_{r - 1} = {}^{n + 1}C_{r} - {}^{n}C_{r}$

So, ${}^{51}C_{3} = {}^{52}C_{4} - {}^{51}C_{4}, {}^{50}C_{3} = {}^{51}C_{4} - {}^{50}C_{4}$ and ${}^{45}C_{3} = {}^{46}C_{4} - {}^{45}C_{4}$

$\Rightarrow$ $\sum_{k = 0}^{6} {}^{51 - k}C_{3} = {}^{52}C_{4} - {}^{51}C_{4} + {}^{51}C_{4} - {}^{50}C_{4} + \dots + {}^{47}C_{4} - {}^{46}C_{4} + {}^{46}C_{4} - {}^{45}C_{4}$

$= {}^{52}C_{4} - {}^{45}C_{4}$

Q 46 :

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ________.

(3483638676)

Required number of ways

$= {}^{3}C_{0} 3^{20} - {}^{3}C_{1} 2^{20} + {}^{3}C_{2} 1^{20}$

$= 3^{20} - 3 \times 2^{20} + 3 = 3 (3^{19} - 2^{20} + 1) = 3483638676$

Q 47 :

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 without repetition is ________.

(432)

Vowels are U, I, E, E and consonants are N, V, R, S.

So, there are 4 vowels and 4 consonants.

When both vowels are different,

Number of ways $= {}^{3}C_{2} \times {}^{4}C_{2} \times 4! = 432$

Q 48 :

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is ________.

(546)

5 language courses and 7 non-language courses.

He can select:

(i) 0 language courses and 5 non-language courses in ${}^{5}C_{0} \times {}^{7}C_{5} = 21$ ways

(ii) 1 language course and 4 non-language courses in ${}^{5}C_{1} \times {}^{7}C_{4} = 175$ ways

(iii) 2 language courses and 3 non-language courses in ${}^{5}C_{2} \times {}^{7}C_{3} = 350$ ways

Required number of ways $= 21 + 175 + 350 = 546$ ways.

Q 49 :

The rank of the word 'SUCCESS' in the dictionary is ________.

(331)

The alphabetical order is C, E, S, U. The number of words beginning with C is $\frac{6!}{3!} = 120$ ways, and those beginning with E is $\frac{6!}{2! 3!} = 60$ ways.

Then come words beginning with SC, numbering $\frac{5!}{2!} = 60$ , SE, numbering $\frac{5!}{2! 2!} = 30$ , and SS, numbering $\frac{5!}{2!} = 60$ .

After which, comes the word SUCCESS.

Thus, the rank of SUCCESS is $120 + 60 + 60 + 30 + 60 + 1 = 331$

Q 50 :

The rank of the word 'ARRANGE' in the dictionary is ________.

(342)

The alphabetical order is A, E, G, N, R. The number of words beginning with AA is $\frac{5!}{2!} = 60$ , beginning with AE is $\frac{5!}{2!} = 60$ , beginning with AG is $\frac{5!}{2!} = 60$ , beginning with AN is $\frac{5!}{2!} = 60$ . Then come words beginning with ARA is $4! = 24$ , beginning with ARE is $4! = 24$ , beginning with ARG is $4! = 24$ , and beginning with ARN is $4! = 24$ , then come words beginning ARRAE is $2! = 2$ , beginning with ARRAG is $2! = 2$ , then next words are ARRANEG and ARRANGE.

Thus, the rank of ARRANGE is $60 + 60 + 60 + 60 + 24 + 24 + 24 + 24 + 2 + 2 + 2 = 342$

Q 51 :

The number of four-letter words that can be formed using the letters of the word BARRACK is ________.

(270)

Given word : BARRACK

Case 1: If all four letters are different, then number of words $= {}^{5}C_{4} \times 4! = 120$

Case 2: If two letters are of same kind and other are different, then number of words $= {}^{2}C_{1} \times {}^{4}C_{2} \times \frac{4!}{2!} = 144$

Case 3: If first two are same and second two are also same letters, then number of words $= \frac{4!}{2! 2!} = 6$

$∴$ Total number of words $= 120 + 144 + 6 = 270$

Q 52 :

Relatives of a man comprise 4 ladies and 3 gentlemen, and his wife has also 7 relatives: 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

(485)

3 gentlemen and 3 ladies may be chosen as follows:

Case	Man	Wife
Case I	3(L)	3(G)
Case II	2(L), 1(G)	2(G), 1(L)
Case III	1(L), 2(G)	2(L), 1(G)
Case IV	3(G)	3(L)

Required number of ways of inviting 3 ladies and 3 gentlemen to the party

$= {}^{4}C_{3} \times {}^{4}C_{3} + ({}^{4}C_{2} \times {}^{3}C_{1}) ({}^{3}C_{1} \times {}^{4}C_{2}) + ({}^{4}C_{1} \times {}^{3}C_{2}) ({}^{3}C_{2} \times {}^{4}C_{1}) + {}^{3}C_{3} \times {}^{3}C_{3}$

$= 16 + 324 + 144 + 1 = 485$

Q 53 :

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 of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $\frac{m}{n}$ is

$\frac{7}{5}$
$\frac{5}{2}$
$\frac{1}{24}$
$5$

1

2

3

4

(4)

$m = {}^{6}C_{2} \times {}^{5}C_{4} \times 2! \times 4! \times 5!$

$n = 5! \times 6!$

Q 54 :

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 ‘MANKIND’ is

1492
1493
1490
1491

1

2

3

4

(1)

$\begin{matrix} M & A & N & K & I & N & D \end{matrix}$

$(\frac{4 \times 6!}{2!}) + (5! \times 0) + (\frac{4! \times 3}{2!}) + (3! \times 2) + (2! \times 1) + (1! \times 1) + (0! \times 0) = 1492$

$\Rightarrow 1440 + 36 + 12 + 4 = 1492$

Q 55 :

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: [2026]

5040
3050
3410
4320

1

2

3

4

(4)

Required ways = Total ways of arrangement – All girls together

= 7! – 5! $\times$ 3! = 5040 – 720= 4320

Q 56 :

There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

840
600
276
640

1

2

3

4

(1)

$∴$ $x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 3 \to (remaining)$

$∴$ ${}^{3 + 5 - 1}{C_{5 - 1}} = {}^{7}{C_{4}} = {}^{7}{C_{3}}$

$∴$ $total arrangement = {}^{7}{C_{3}} \times 4! = 840$

Q 57 :

Number of permutations of the word IITJEE is

(180)

$\frac{6!}{2! 2!}$

Q 58 :

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

89
100
258
237

1

2

3

4

(1)

We arrange the letters of OUGHT in alphabetical order as G,H,O,T,U

In dictionary words starting with

$G \cdot \cdot \cdot \cdot \cdot \cdot \cdot \to 4!$

$H \cdot \cdot \cdot \cdot \cdot \cdot \cdot \to 4!$

$O \cdot \cdot \cdot \cdot \cdot \cdot \cdot \to 4!$

$T G \cdot \cdot \cdot \cdot \cdot \cdot \to 3!$

$T H \cdot \cdot \cdot \cdot \cdot \cdot \to 3!$

$T O G \cdot \cdot \cdot \cdot \cdot \to 2!$

$T O H \cdot \cdot \cdot \cdot \cdot \to 2!$

$\frac{T O U G H \to 1!}{Total=89}$

Q 59 :

The number of 5 digit numbers, greater than 70000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is _____

120
168
220
48

1

2

3

4

(4)

Topic Question Set

Q 41 :

The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ________.

Q 42 :

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 distinct and two letters are alike, is ________.

Q 43 :

The value of ${}^{16}C_{9} + {}^{16}C_{10} - {}^{16}C_{6} - {}^{16}C_{7}$ is

Q 44 :

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicket keepers. The number of ways a team of 11 players can be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicket keeper is ________.

Q 45 :

The value of $\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to

Q 46 :

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ________.

Q 47 :

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 without repetition is ________.

Q 48 :

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is ________.

Q 49 :

The rank of the word 'SUCCESS' in the dictionary is ________.

Q 50 :

The rank of the word 'ARRANGE' in the dictionary is ________.

Q 51 :

The number of four-letter words that can be formed using the letters of the word BARRACK is ________.

Q 52 :

Relatives of a man comprise 4 ladies and 3 gentlemen, and his wife has also 7 relatives: 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

Q 54 :

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 ‘MANKIND’ is

Q 55 :

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: [2026]

Q 56 :

There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

Q 57 :

Number of permutations of the word IITJEE is

Q 58 :

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

Q 59 :

The number of 5 digit numbers, greater than 70000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is _____

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Topic Question Set

Q 41 : The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ________.

Q 42 : 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 distinct and two letters are alike, is ________.

Q 43 : The value of C916+C1016-C616-C716 is

Q 44 : There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicket keepers. The number of ways a team of 11 players can be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicket keeper is ________.

Q 45 : The value of ∑k=06C351-k is equal to

Q 46 : The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ________.

Q 47 : 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 without repetition is ________.

Q 48 : A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is ________.

Q 49 : The rank of the word 'SUCCESS' in the dictionary is ________.

Q 50 : The rank of the word 'ARRANGE' in the dictionary is ________.

Q 51 : The number of four-letter words that can be formed using the letters of the word BARRACK is ________.

Q 52 : Relatives of a man comprise 4 ladies and 3 gentlemen, and his wife has also 7 relatives: 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

Q 54 : 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 ‘MANKIND’ is

Q 55 : The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: [2026]

Q 56 : There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

Q 57 : Number of permutations of the word IITJEE is

Q 58 : 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

Q 59 : The number of 5 digit numbers, greater than 70000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is _____

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Q 41 :

The number of 4-letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is ________.

Q 42 :

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 distinct and two letters are alike, is ________.

Q 43 :

The value of ${}^{16}C_{9} + {}^{16}C_{10} - {}^{16}C_{6} - {}^{16}C_{7}$ is

Q 44 :

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicket keepers. The number of ways a team of 11 players can be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicket keeper is ________.

Q 45 :

The value of $\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to

Q 46 :

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ________.

Q 47 :

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 without repetition is ________.

Q 48 :

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is ________.

Q 49 :

The rank of the word 'SUCCESS' in the dictionary is ________.

Q 50 :

The rank of the word 'ARRANGE' in the dictionary is ________.

Q 51 :

The number of four-letter words that can be formed using the letters of the word BARRACK is ________.

Q 52 :

Relatives of a man comprise 4 ladies and 3 gentlemen, and his wife has also 7 relatives: 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

Q 54 :

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 ‘MANKIND’ is

Q 55 :

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: [2026]

Q 56 :

There are 10 seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is

Q 57 :

Number of permutations of the word IITJEE is

Q 58 :

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

Q 59 :

The number of 5 digit numbers, greater than 70000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is _____