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

Q 1 :
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

1

2

3

4

(A)

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 2 :
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

1

2

3

4

(A)

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 3 :
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

1

2

3

4

(D)

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 4 :
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 $β \notin N$

1

2

3

4

(C)

Q 5 :
Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

18
15
12
16

1

2

3

4

(B)

Ways of arranging 8 identical books in 4 identical shelves = {8, 0, 0, 0}, {7, 1, 0, 0}, {6, 2, 0, 0}, {5, 3, 0, 0}, {4, 4, 0, 0}, {6, 1, 1, 0},{5, 2, 1, 0}, {4, 3, 1, 0}, {4, 2, 2, 0}, {3, 3, 2, 0}, {5, 1, 1, 1}, {4, 2, 1, 1}, {3, 3, 1, 1}, {3, 2, 2, 1}, {2, 2, 2, 2}.

Number of ways = 15

[As shelves and books are identical, so the arrangements (5, 2, 1, 0), (2, 5, 1, 0), and (1, 2, 5, 0) are considered as same]

Q 6 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

(125)

We need sum of 16 on throwing a dice four times so possible ways are:
{(4, 4, 4, 4), (5, 4, 4, 3), (5, 5, 1, 5), (5, 5, 3, 3), (5, 5, 4, 2), (6, 4, 3, 3), (6, 4, 4, 2), (6, 5, 3, 2), (6, 5, 4, 1), (6, 6, 3, 1), (6, 6, 2, 2)}

Obtained Result	Number of Ways
(6, 6, 2, 2)	$\frac{4!}{2! 2!} = 6$
(6, 6, 3, 1)	$\frac{4!}{2!} = 12$
(6, 5, 4, 1)	$4! = 24$
(6, 5, 3, 2)	$4! = 24$
(6, 4, 4, 2)	$\frac{4!}{2!} = 12$
(6, 4, 3, 3)	$\frac{4!}{2!} = 12$
(5, 5, 4, 2)	$\frac{4!}{2!} = 12$
(5, 5, 3, 3)	$\frac{4!}{2! 2!} = 6$
(5, 5, 1, 5)	$\frac{4!}{3!} = 4$
(5, 4, 4, 3)	$\frac{4!}{2!} = 12$
(4, 4, 4, 4)	$1! = 1$
Total ways	125

Q 7 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

(70)

Number between 100 to 1000 are 3-digit number.

Let number $a b c$ such that $a + b + c = 14$

where $a, b, c \in {0, 1, 2, . . . 9}$ and $a \geq 1$

Case I : All three digits are same ie., $a = b = c$

$\Rightarrow 3 a = 14$ , which is not possible

Case II : Two digits are same i.e., $a = b \neq c$

$\Rightarrow 2 a + c = 14$

$∴$ (a, c) = {(3, 8), (4, 6), (5, 4), (6, 2), (7, 0)}

In each subcase, number of ways of forming a 3-digit number = $\frac{3!}{2!}$

There are 5 such cases as (a, c) have 5 elements in which 0,7,7 is included

So, total number of 3 digits numbers when 2 digits are same $5 \times \frac{3!}{2!} - 1$

$= 15 - 1 = 14$

Case III : All digits are different

$∴$ (a, b, c) = {(1, 4, 9), (2, 4, 8), (2, 3, 9), (1, 5, 8), (3, 4, 7), (2, 5, 7), (1, 6, 7), (3, 5, 6), (5, 9, 0), (6, 8, 0)}

Total number of 3 digits number formed by above triplet = $10 \times 3! - 2 \times 2! = 60 - 4 = 56$

Total number of required number = 56 + 14 = 70

Q 8 :
All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

0%

(553)

Number of words starts with $E = \frac{6!}{2!}$

Number of words starts with $G E = \frac{5!}{2!}$

Number of words starts with $G N = \frac{5!}{2!}$

Number of words starts with $GTE = 4!$

Number of words starts with $GTN = 4!$

Number of words starts with $GTT = 4!$

Next word will be GTWENTY

$∴$ Rank of GTWENTY $= \frac{6!}{2!} + 2 \times \frac{5!}{2!} + 3 \times 4! + 1 = 553$

Topic Question Set

Q 1 :
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 2 :
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 3 :
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 4 :
Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

(C)

Q 5 :
Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 6 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

Q 7 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

Q 8 :
All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

0%

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Topic Question Set

Q 1 : 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 2 : 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 3 : 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 4 : Let α=(4!)!(4!)3! and β=(5!)!(5!)4!. Then

(C)

Q 5 : Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 6 : The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

0%

Q 7 : The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

0%

Q 8 : All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]

0%

Want Us to Email you About Special Offers & Updates?

Q 1 :
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 2 :
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 3 :
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 4 :
Let $α = \frac{(4!)!}{{(4!)}^{3!}}$ and $β = \frac{(5!)!}{{(5!)}^{4!}} .$ Then

Q 5 :
Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to [2024]

Q 6 :
The number of ways of getting a sum 16 on throwing a dice four times is ______. [2024]

Q 7 :
The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ______. [2024]

Q 8 :
All the letters of the word “GTWENTY” are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word “GTWENTY” is _____. [2024]