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

Q 1 :
All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is [2023]

580
582
576
578

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4

(2)

No. of words formed when word starts from B = 5!

No. of words formed when word starts from C = 5!

No. of words formed when word starts from I = 5!

No. of words formed when word starts from L = 5!

No. of words formed when word starts from PB = 4!

No. of words formed when word starts from PC = 4!

No. of words formed when word starts from PI = 4!

No. of words formed when word starts from PL = 4!

No. of words formed when word starts from PUBC = 2!

No. of words formed when word starts from PUBI = 2!

No. of words formed when word starts from PUBLC = 1

Now, the word comes is PUBLIC = 1

∴ Serial number of word PUBLIC

= 4 × 5! + 4 × 4! + 2 × 2! + 1 + 1 = 582

Q 2 :
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

14800
16800
18000
33600

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4

(2)

Q 3 :
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 [2023]

$7 {(360)}^{2}$
$126 {(5!)}^{2}$
$7 {(720)}^{2}$
$720$

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(2)

[IMAGE 17]

Number of girls = 5

Number of boys = 7

The number of ways of arranging boys around a table is (7 – 1)! = 6!

Now, there are 7 spaces in between two boys, so 5 girls arranged in 7 gaps by ${}^{7}{P_{5}}$ ways.

So, required number of ways = $6! \times {}^{7}{P_{5}} = 126 \times {(5!)}^{2}$

Q 4 :
If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to [2023]

5670
945
2835
1890

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4

(1)

In the word ''MATHEMATICS'' there are, $2 \to M's; 2 \to A's; 2 \to T's$

Total number of words formed by the letters of the word ''MATHEMATICS'' is $\frac{⌊ 11}{⌊ 2 \times ⌊ 2 \times ⌊ 2}$

Number of words in which 'C' and 'S' come together is $\frac{⌊ 10}{⌊ 2 \times ⌊ 2 \times ⌊ 2} \times 2$

Then, number of words in which 'C' and 'S' do not come together is $\frac{⌊ 11}{⌊ 2 \times ⌊ 2 \times ⌊ 2} - \frac{2 ⌊ 10}{⌊ 2 \times ⌊ 2 \times ⌊ 2} = \frac{9 \times ⌊ 10}{2 \times 2 \times 2}$

$\Rightarrow \frac{9 \times 10 \times 9 \times 8 \times 7 \times 6!}{2 \times 2 \times 2} = 6! \times k (∵ Given) \Rightarrow k = 5670$

Q 5 :
Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is [2023]

3360
1120
1680
560

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(3)

8 persons can be divided into 3 groups of 3, 3, and 2 members each.

$∴$ $Required number of ways = \frac{8!}{3! 3! 2! 3!} \times 3!$

$= \frac{8 \times 7 \times 6 \times 5 \times 4}{4} = 56 \times 30 = 1680$

Q 6 :
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 THAMS is [2023]

103
104
101
102

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4

(1)

Given word is MATHS.

In alphabetical order: A, H, M, S, T

Now, word starting with

A_ _ _ _ i.e., 4! ; H_ _ _ _ i.e., 4!

M_ _ _ _ i.e., 4! ; S_ _ _ _ i.e., 4!

T A_ _ _ i.e., 3! ; T H A M S i.e., 1

$∴$ $Rank of the word THAMS = 4 \times 4! + 1 \times 3! + 1 = 103$

Q 7 :
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

NRAPGU
NRAGPU
NRAPUG
NRAGUP

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(1)

Letter in the word NAGPUR = 6

If we fix A _ _ _ _ _

Number of words start with A = 5! = 120

If we fix G _ _ _ _ _

Number of words start with G = 5! = 120

If we fix N A _ _ _ _

Number of words start with NA = 4! = 24

If we fix N G _ _ _ _

Number of words start with NG = 4! = 24

If we fix N P _ _ _ _

Number of words start with NP = 4! = 24

Next word i.e., $313^{th}$ word in dictionary will be NRAGPU, $314^{th}$ word will be NRAGUP, $315^{th}$ word will be NRAPGU.

Q 8 :
60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

OBBJH
HBBJO
OBBHJ
JBBOH

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(1)

Let us fix B first (Dictionary Order)

B _ _ _ _ (We have 4 distinct letters left, to be arranged in 4 distinct ways)
Number of ways = 4! = 24

H _ _ _ _ (We have B, B, J, O to be arranged in 4 ways when H is fixed)

Number of ways = $\frac{4!}{2!} = 12$

J _ _ _ _ (B, B, H, O to be arranged in 4 ways)

Number of ways = $\frac{4!}{2!} = 12$

49th word will be OBBHJ
50th word will be OBBJH

Q 9 :
If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

43
47
53
51

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(4)

Number of ways in which five different employees can sit are

$I$ $I I$ $I I I$ $I V$

$□$ $□$ $□$ $□$

5 0 0 0 $\to$ 5!/5! = 1

4 1 0 0 $\to$ 5!/4! = 5

3 2 0 0 $\to$ 5!/3!2! = 10

3 1 1 0 $\to$ 5!/3! 1! 1! 2! = 10

2 2 1 0 $\to$ $\frac{5!}{2! 2! 1! 2!} = 15$

2 1 1 1 $\to$ $\frac{5!}{2! 1! 1! 1! 3!} = 10$

$∴$ Total number of ways =1 + 5 + 10 + 10 + 15 + 10 = 51

Q 10 :
Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

$α \notin N$ and $β \in N$
$α \in N$ and $β \notin N$
$α \in N$ and $β \in N$
$α \notin N$ and $β \notin N$

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Topic Question Set

Q 1 :
All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is [2023]

Q 2 :
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

(2)

Q 3 :
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 [2023]

Q 4 :
If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to [2023]

Q 5 :
Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is [2023]

Q 6 :
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 THAMS is [2023]

Q 7 :
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

Q 8 :
60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 9 :
If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

Q 10 :
Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

(3)

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Topic Question Set

Q 1 : All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is [2023]

Q 2 : The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

(2)

Q 3 : 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 [2023]

Q 4 : If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to [2023]

Q 5 : Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is [2023]

Q 6 : 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 THAMS is [2023]

Q 7 : If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315th position in this arrangement is [2024]

Q 8 : 60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 9 : If n is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to [2024]

Q 10 : Let α=(4!)!(4!)3! and β=(5!)!(5!)4!. Then

(3)

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Q 1 :
All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is [2023]

Q 2 :
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

Q 3 :
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 [2023]

Q 4 :
If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!)k, then k is equal to [2023]

Q 5 :
Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is [2023]

Q 6 :
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 THAMS is [2023]

Q 7 :
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is [2024]

Q 8 :
60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50th word is [2024]

Q 9 :
If $n$ is the number of ways in which five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to [2024]

Q 10 :
Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then