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

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

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

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?

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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?

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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?