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

1

2

3

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

1

2

3

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]

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

1

2

3

4

(2)

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

1

2

3

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

1

2

3

4

(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

1

2

3

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 :

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to [2023]

132
120
72
96

1

2

3

4

(2)

Numbers greater than 40000 and divisible by 5 can be formed using digits 0, 1, 3, 5, 7, 9 in the following ways:

	Number of ways
5 _ _ _ 0	4 × 3 × 2 × 1 = 24
7 _ _ _ 0
7 _ _ _ 5	4 × 3 × 2 × 2 = 48
9 _ _ _ 0
9 _ _ _ 5	4 × 3 × 2 × 2 = 48

Total number of ways = 24 + 48 + 48 = 120

Q 8 :

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is [2023]

327
328
326
324

1

2

3

4

(1)

Given, A, D, M, N, O, Y

A → 5! ; D → 5! ; M A → 4! ; M N → 4!

M O A → 3! ; M O D → 3! ; M O N A → 2! ; M O N D A Y

⇒ Rank = 2(5)! + 3(4)! + 2(3)! + 2! + 1

= 240 + 72 + 12 + 2 + 1 = 327

Q 9 :

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is [2023]

18
21
22
20

1

2

3

4

(3)

Case I: All the three digits are same i.e., 111, 333, 555, 888. All the numbers are divisible by 3.

Case II: Two digits are same.

We can see that when two digits are same, then numbers having digits 5, 5, 8 or 8, 8, 5 will be divisible by 3 as their sum is divisible by 3.

Now, for each number, we have $\frac{3!}{2!} = 3$ arrangements.

$∴$ In this case, we have 6 numbers divisible by 3.

Case III: All digits are distinct.

i.e., (1, 3, 5), (1, 3, 8)

So, number formed = 2 × 3! = 12

Hence, total 3-digit numbers formed by digits 1, 3, 5, 8 and divisible by 3 is 4 + 6 + 12 = 22.

Q 10 :

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is [2023]

225
120
125
150

1

2

3

4

(2)

In first column, 1 can be placed in any of the 5 places = 5

In second column, 1 can be placed in any of the 4 places = 4

In third column, 1 can be placed in any of the 3 places = 3

In fourth column, 1 can be placed in any of the 2 places = 2

In fifth column, 1 can be placed in any of the 1 place = 1

Required number of ways = 5 × 4 × 3 × 2 × 1 = 120

Q 11 :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is [2023]

168
120
220
48

1

2

3

4

(1)

5 digits numbers = 5! = 120

Four digit number greater than 7000 = 2 × 4 × 3 × 2 = 48

Total number greater than 7000 = 120 + 48 = 168

Q 12 :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is [2023]

72
120
6
12

1

2

3

4

(1)

Number of numbers lies between 5000 and 10000 are 4 digit numbers greater than 5000 so, the first digit can be filled in 3 ways (using 5, 7, 9) and as repetition is not allowed, the other choices are 4, 3, 2.

Total number of number = 3 × 4 × 3 × 2 = 72

Q 13 :

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 [2023]

86
89
79
84

1

2

3

4

(2)

The letters in sequence of the word ‘OUGHT’ in alphabetical order as follows: G, H, O, T, U
Word starts with G can arrange in 4! ways.
H can arrange in 4! ways.
O can arrange in 4! ways.
TG can arrange in 3! ways.
TH can arrange in 3! ways.
TOG can arrange in 2! ways.
TOH can arrange in 2! ways.
TOUGH can arrange in 1! way.

So we must have serial number of the word is 89.

Q 14 :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is [2023]

400
507
432
472

1

2

3

4

(3)

We know that we have total 3-digit numbers only 900.

$∴$ $Total 3 digit numbers = 900$

Out of these 900, the number divisible by 3 $= 300$ $(∵ \frac{900}{3} = 300)$

In the same way, the numbers divisible by 4 $= \frac{900}{4} = 225$

So, divisible by 3 and 4 = 75 $(Using \frac{900}{12} = 75)$

Number divisible by either 3 or 4 $= 300 + 225 - 75 = 450$

Now, we have to remove the numbers divisible by 48, 144, 192 ....... 18 terms.

Hence, required numbers, who are divisible by either 3 or 4 but not by 48 is $450 - 18 = 432$

Q 15 :

The number of ways of selecting two numbers $a$ and $b, a \in {2, 4, 6 . . ., 100}$ and $b \in {1, 3, 5 . . ., 99}$ such that 2 is the remainder when $a + b$ is divided by 23 is [2023]

54
108
268
186

1

2

3

4

(2)

$a \in {2, 4, 6, \dots, 100}, b \in {1, 3, 5, \dots, 99}$

$a + b = 23 λ + 2$

$λ = 0, 1, 2, \dots, 22 but λ can't be even.$

$If λ = 1, (a, b) \to 12 pairs;$ $λ = 3, (a, b) \to 35 pairs;$

$λ = 5, (a, b) \to 42 pairs;$ $λ = 7, (a, b) \to 19 pairs;$

$λ = 9, (a, b) \to 0 pairs.$

$∴$ $Total number of ways = 12 + 35 + 42 + 19 = 108$

Q 16 :

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ________ . [2023]

(4898)

Given digits are 1, 2, 3, 4, 5, 6, 7

Total number of permutations = 7!

Let A = number of numbers containing string 153

Let B = number of numbers containing string 2467

$n (A) = 5! \times 1, n (B) = 4! \times 1, n (A \cap B) = 2!$

Required number of permutations = Total number of permutations - $n (A \cup B)$

$= 7! - (5! + 4! - 2!) = 7! - 142 = 4898$

Q 17 :

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _________ . [2023]

(26664)

Sum of all unit place numbers

$= 3! \times 2 + \frac{3!}{2!} \times 3 + \frac{3!}{2!} \times 1 = 12 + 9 + 3 = 24$

$∴$ $Sum of all 4 digit numbers:$

$= 1 \times 24 + 10 \times 24 + 100 \times 24 + 1000 \times 24$

$= 24000 + 2400 + 240 + 24 = 26, 664$

Q 18 :

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is __________ . [2023]

(44)

Dearrangement of 5 students

$D_{5} = 5! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!})$

$= 120 (\frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120}) = 60 - 20 + 5 - 1 = 44$

Q 19 :

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ . [2023]

(413)

Case 1: $1 1 1 1 1 4 3$

Number of ways = $\frac{7!}{5!} = 42$ [Since, 1 repeats five times]

Case 2: $1 1 1 1 3 3 2$

Number of ways = $\frac{7!}{4! 2!} = 105$ [Since, 1 repeats 4 times and 3 repeats 2 times]

Case 3: $2 2 2 3 1 11$

Number of ways = $\frac{7!}{3! 3!} = 140$ [Since, 2 repeats 3 times and 1 repeats 3 times]

Case 4: $22 2 2 21 1$

Number of ways = $\frac{7!}{5! 2!} = 21$ [Since, 2 repeats 5 times and 1 repeats 2 times]

Case 5: $1 11 14 2 2$

Number of ways = $\frac{7!}{4! 2!} = 105$ [Since, 1 repeats 4 times and 2 repeats 2 times]

$∴$ $Total number of ways = 42 + 105 + 140 + 21 + 105 = 413$

Q 20 :

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________ . [2023]

(16)

Total 3-digit numbers formed by 1, 2, 3, 4, 5 when the last digit in 4 and divisible by 6.

$\begin{matrix} 1 & 1 & \to & 1 \\ 4 & 4 & \to & 1 \\ 1 & 4 & \to & 2! \\ 2 & 3 & \to & 2! \\ 3 & 5 & \to & 2! \end{matrix}] = 8$

When last digit is 2:

$\begin{matrix} 2 & 2 & \to & 1 \\ 5 & 5 & \to & 1 \\ 1 & 3 & \to & 2! \\ 2 & 5 & \to & 2! \\ 4 & 3 & \to & 2! \end{matrix}] = 8$

$∴ Total 3-digit numbers formed by 1, 2, 3, 4, 5 which are divisible by 6 are 16 .$

Q 21 :

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits.Then the maximum number of trials necessary to obtain the correct code is _____ . [2023]

(72)

Sum of first two digits

Sum of last two digits = $α$

Case–I : $α$ = 7

2 × 12 = 24 ways.

2 × 8 ways = 16 ways

2 × 4 ways = 8 ways

Total ways = 24 + 16 + 16 + 8 + 8 = 72

Q 22 :

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ___________ . [2023]

(514)

Number of numbers divisible by 2 = 450
Number of numbers divisible by 3 = 300
Number of numbers divisible by 7 = 128
Number of numbers divisible by 2 & 7 = 64
Number of numbers divisible by 3 & 7 = 43
Number of numbers divisible by 2 & 3 = 150
Number of numbers divisible by 2, 3 & 7 = 21

Total numbers = 450 + 300 – 150 – 64 – 43 + 21 = 514

Q 23 :

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 __________ . [2023]

(50400)

In the word ASSASSINATION, there are 6 vowels A A A I I O and consonants S S S S N N T.

$Taking (AAAIIO) as a single unit, we get:$

$Number of ways to arrange ASSASSINATION such that all vowels occur together$

$= \frac{8!}{4! 2!} \times \frac{6!}{3! 2!} = \frac{8! \times 6!}{4! {(2!)}^{2} 3!} = 50, 400$

Q 24 :

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

(81)

Total number of 6-digit numbers formed by the digits 4, 5, 9 and divisible by 6 is when all the digits are same.

$444444 \to \frac{6!}{6!} = 1$

$Taking two digits (4, 5), 555444 \to \frac{5!}{3! 2!} = 10$

$For (4, 9), 999444 \to \frac{5!}{3! 2!} = 10$

$Taking three digits: 459444 \to \frac{5!}{3!} = 20$

$559554 \to \frac{5!}{4!} = 5, 959994 \to \frac{5!}{4!} = 5$

$959454 \to \frac{5!}{2! 2!} = 30$

$Total number = 20 + 5 + 5 + 30 + 10 + 10 + 1 = 81$

Q 25 :

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 __________ . [2023]

(60)

Even numbers are 2, 4, 2, 4

Odd numbers are 1, 3, 1, 3, 1

$\bar{O} \bar{E} \bar{O} \bar{E} \bar{O} \bar{E} \bar{O} \bar{E} \bar{O}$

$Even places can be filled with even numbers in \frac{4!}{2! 2!} ways.$

$Similarly, odd places can be filled in \frac{5!}{2! 3!} ways.$

$Required number of ways = \frac{4!}{2! 2!} \times \frac{5!}{2! 3!} = 60$

Q 26 :

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25 .$ Then, the number of ways of choosing $x$ and $y,$ such that $x + y$ is divisible by 5, is _________ . [2023]

(120)

Let $x + y = 5 λ$

Possible cases are

$\begin{matrix} x & y & Number of ways \\ 5 λ & 5 λ & 20 \\ 5 λ + 1 & 5 λ + 4 & 25 \\ 5 λ + 2 & 5 λ + 3 & 25 \\ 5 λ + 3 & 5 λ + 2 & 25 \\ 5 λ + 4 & 5 λ + 1 & 25 \end{matrix}$

$Total number of ways = 20 + 25 + 25 + 25 + 25 = 120$

Q 27 :

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 serial number 1. Then the serial number of 35337 is _______ . [2023]

(1436)

$Number of 5-digit numbers starting with digit 1 = 5 \times 5 \times 5 \times 5 = 625$

$No. of 5-digit numbers starting with digit 2 = 5^{4} = 625$

$No. of 5-digit numbers starting with 31 = 5 \times 5 \times 5 = 125$

$No. of 5-digit numbers starting with 32 = 5 \times 5 \times 5 = 125$

$No. of 5-digit numbers starting with 33 = 5 \times 5 \times 5 = 125$

$No. of 5-digit numbers starting with 351 = 5 \times 5 = 25$

$No. of 5-digit numbers starting with 352 = 5 \times 5 = 25$

$No. of 5-digit numbers starting with 3531 = 5$

$No. of 5-digit numbers starting with 3532 = 5$

$Before 35337, there will be 4 numbers, so the rank of 35337 will be 1690.$

$So, in descending order serial number will be,$ $3125 - 1690 + 1 = 1436$

Q 28 :

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _________ . [2023]

(3000)

$The total 4-digit numbers is N$

$= 9 \times 10 \times 10 \times 10 = 9000$

$The number N should be divisible by 2 but not by 3.$

$N = (Number divisible by 2) - (Number divisible by 6)$

$= \frac{9000}{2} - \frac{9000}{6} = 4500 - 1500 = 3000$

Q 29 :

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ . [2023]

(21)

$We have to make 4-digit numbers using the digits 1, 2, 3, and 5 .$

$The unit digit of the 4-digit number will be 5 .$

x

y

z

5

$Now, the sum (x + y + z) should be of the form (3 λ + 1) .$

$Therefore, the possible cases are:$

$(x, y, z) = (1, 1, 5), (1, 1, 2), (2, 2, 3), (2, 3, 5), (3, 3, 1), (5, 5, 3)$

$So, total arrangements are:$

$For (1, 1, 5) \to \frac{3!}{2!} = 3; For (1, 1, 2) \to \frac{3!}{2!} = 3$

$For (2, 2, 3) \to \frac{3!}{2!} = 3; For (2, 3, 5) \to 3! = 6$

$For (3, 3, 1) \to \frac{3!}{2!} = 3; For (5, 5, 3) \to \frac{3!}{2!} = 3$

$So, total number of arrangements = 3 + 3 + 3 + 6 + 3 + 3 = 21$

Q 30 :

Let $\sum_{n = 0}^{\infty} \frac{n^{3} ((2 n)!) + (2 n - 1) (n!)}{(n!) ((2 n)!)} = a e + \frac{b}{e} + c,$ where $a, b, c \in ℤ$ and $e = \sum_{n = 0}^{\infty} \frac{1}{n!}$ . Then $a^{2} - b + c$ is equal to ________ . [2023]

(26)

We have, $\sum_{n = 0}^{\infty} \frac{n^{3} (2 n!) + (2 n - 1) (n!)}{(n!) ((2 n)!)} = a e + \frac{b}{e} + c$

$\Rightarrow \sum_{n = 0}^{\infty} \frac{1}{(n - 3)!} + \sum_{n = 0}^{\infty} \frac{3}{(n - 2)!} + \sum_{n = 0}^{\infty} \frac{1}{(n - 1)!} + \sum_{n = 0}^{\infty} \frac{1}{(2 n - 1)!} - \sum_{n = 0}^{\infty} \frac{1}{(2 n)!} = a e + \frac{b}{e} + c$

$\Rightarrow e + 3 e + e + \frac{1}{2} (e - \frac{1}{e}) - \frac{1}{2} (e + \frac{1}{e}) = a e + \frac{b}{e} + c$

$\Rightarrow 5 e - \frac{1}{e} = a e + \frac{b}{e} + c$

$Comparing, we get, a = 5, b = - 1, c = 0$

$a^{2} - b + c = 25 + 1 = 26$

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 :

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to [2023]

Q 8 :

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is [2023]

Q 9 :

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is [2023]

Q 10 :

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is [2023]

Q 11 :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is [2023]

Q 12 :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is [2023]

Q 13 :

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 [2023]

Q 14 :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is [2023]

Q 15 :

The number of ways of selecting two numbers $a$ and $b, a \in {2, 4, 6 . . ., 100}$ and $b \in {1, 3, 5 . . ., 99}$ such that 2 is the remainder when $a + b$ is divided by 23 is [2023]

Q 16 :

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ________ . [2023]

Q 17 :

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _________ . [2023]

Q 18 :

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is __________ . [2023]

Q 19 :

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ . [2023]

Q 20 :

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________ . [2023]

Q 22 :

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ___________ . [2023]

Q 23 :

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 __________ . [2023]

Q 24 :

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

Q 25 :

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 __________ . [2023]

Q 26 :

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25 .$ Then, the number of ways of choosing $x$ and $y,$ such that $x + y$ is divisible by 5, is _________ . [2023]

Q 27 :

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 serial number 1. Then the serial number of 35337 is _______ . [2023]

Q 28 :

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _________ . [2023]

Q 29 :

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ . [2023]

Q 30 :

Let $\sum_{n = 0}^{\infty} \frac{n^{3} ((2 n)!) + (2 n - 1) (n!)}{(n!) ((2 n)!)} = a e + \frac{b}{e} + c,$ where $a, b, c \in ℤ$ and $e = \sum_{n = 0}^{\infty} \frac{1}{n!}$ . Then $a^{2} - b + c$ is equal to ________ . [2023]

Want Us to Email you About Special Offers & Updates?

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 : The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to [2023]

Q 8 : All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is [2023]

Q 9 : The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is [2023]

Q 10 : The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is [2023]

Q 11 : The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is [2023]

Q 12 : The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is [2023]

Q 13 : 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 [2023]

Q 14 : The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is [2023]

Q 15 : The number of ways of selecting two numbers a and b,a∈{2,4,6...,100} and b∈{1,3,5...,99} such that 2 is the remainder when a+b is divided by 23 is [2023]

Q 16 : The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ________ . [2023]

Q 17 : The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _________ . [2023]

Q 18 : In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is __________ . [2023]

Q 19 : The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ . [2023]

Q 20 : Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________ . [2023]

Q 22 : The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ___________ . [2023]

Q 23 : 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 __________ . [2023]

Q 24 : The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

Q 25 : 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 __________ . [2023]

Q 26 : Let x and y be distinct integers where 1≤x≤25 and 1≤y≤25.Then, the number of ways of choosing x and y, such that x+y is divisible by 5, is _________ . [2023]

Q 27 : 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 serial number 1. Then the serial number of 35337 is _______ . [2023]

Q 28 : The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _________ . [2023]

Q 29 : Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ . [2023]

Q 30 : Let ∑n=0∞n3((2n)!)+(2n-1)(n!)(n!)((2n)!)=ae+be+c, where a,b,c∈ℤ and e=∑n=0∞1n!. Then a2-b+c is equal to ________ . [2023]

Want Us to Email you About Special Offers & Updates?

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 :

The number of five digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to [2023]

Q 8 :

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is [2023]

Q 9 :

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is [2023]

Q 10 :

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is [2023]

Q 11 :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is [2023]

Q 12 :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is [2023]

Q 13 :

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 [2023]

Q 14 :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is [2023]

Q 15 :

The number of ways of selecting two numbers $a$ and $b, a \in {2, 4, 6 . . ., 100}$ and $b \in {1, 3, 5 . . ., 99}$ such that 2 is the remainder when $a + b$ is divided by 23 is [2023]

Q 16 :

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ________ . [2023]

Q 17 :

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to _________ . [2023]

Q 18 :

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is __________ . [2023]

Q 19 :

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is __________ . [2023]

Q 20 :

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________ . [2023]

Q 22 :

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ___________ . [2023]

Q 23 :

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 __________ . [2023]

Q 24 :

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ________ . [2023]

Q 25 :

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 __________ . [2023]

Q 26 :

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25 .$ Then, the number of ways of choosing $x$ and $y,$ such that $x + y$ is divisible by 5, is _________ . [2023]

Q 27 :

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 serial number 1. Then the serial number of 35337 is _______ . [2023]

Q 28 :

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _________ . [2023]

Q 29 :

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ . [2023]

Q 30 :

Let $\sum_{n = 0}^{\infty} \frac{n^{3} ((2 n)!) + (2 n - 1) (n!)}{(n!) ((2 n)!)} = a e + \frac{b}{e} + c,$ where $a, b, c \in ℤ$ and $e = \sum_{n = 0}^{\infty} \frac{1}{n!}$ . Then $a^{2} - b + c$ is equal to ________ . [2023]