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

Q 31 :

If $\frac{1}{{}^{4}C_{n}} = \frac{1}{{}^{5}C_{n}} + \frac{1}{{}^{6}C_{n}}$ , then $n =$

3
2
1
0

1

2

3

4

(2)

$\frac{1}{{}^{4}C_{n}} = \frac{1}{{}^{5}C_{n}} + \frac{1}{{}^{6}C_{n}}$ ; $\frac{1}{{}^{4}C_{n}} = \frac{{}^{6}C_{n} + {}^{5}C_{n}}{{}^{5}C_{n} {}^{6}C_{n}}$

$\Rightarrow {}^{5}C_{n} {}^{6}C_{n} = {}^{4}C_{n} [{}^{6}C_{n} + {}^{5}C_{n}]$

$\Rightarrow \frac{5! 6!}{(5 - n)! (6 - n)! {(n!)}^{2}} = \frac{4!}{(4 - n)! n!} [{}^{6}C_{n} + {}^{5}C_{n}]$

$\Rightarrow \frac{5 \cdot 6}{(5 - n) (6 - n)} = \frac{6}{6 - n} + 1 = \frac{12 - n}{6 - n}$

$\Rightarrow (5 - n) (12 - n) = 30$ $\Rightarrow n^{2} - 17 n + 60 = 30$

$\Rightarrow n^{2} - 17 n + 30 = 0$

$∴$ $n = 15, 2$ $[15 is not possible (∵ n \leq 4)]$

$∴$ $n = 2$

Q 32 :

${}^{2 n}P_{n}$ is equal to

$(n + 1)! \times ({}^{2 n}C_{n})$
$n! \times ({}^{2 n}C_{n})$
$n! \times ({}^{2 n + 1}C_{n})$
$n! \times ({}^{2 n + 1}C_{n + 1})$

1

2

3

4

(2)

${}^{2 n}P_{n} = \frac{(2 n)!}{n!} = n! \times \frac{(2 n)!}{n! n!} = n! \times {}^{2 n}C_{n}$

Q 33 :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

$7 \cdot {}^{6}C_{4} \cdot {}^{8}C_{4}$
$8 \cdot {}^{6}C_{4} \cdot {}^{7}C_{4}$
$6 \cdot 7 \cdot {}^{8}C_{4}$
$6 \cdot 8 \cdot {}^{7}C_{4}$

1

2

3

4

(1)

Leaving S, we have 7 letters $M, I, I, I, P, P, I$

Ways of arranging them $= \frac{7!}{2! 4!} = 7 \cdot 5 \cdot 3$

And four S can be put in 8 places in ${}^{8}C_{4}$ ways.

Therefore, the required number of ways is $= 7 \cdot 5 \cdot 3 \cdot {}^{8}C_{4} = 7 \cdot {}^{6}C_{4} \cdot {}^{8}C_{4}$

Q 34 :

A man has 6 friends. No. of different ways he can invite 2 or more for a dinner is

56
72
28
57

1

2

3

4

(4)

$Required number = {}^{6}C_{2} + {}^{6}C_{3} + {}^{6}C_{4} + {}^{6}C_{5} + {}^{6}C_{6}$

$= 15 + 20 + 15 + 6 + 1 = 57$

Q 35 :

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

64
24
3
12

1

2

3

4

(1)

$In order to draw 3 balls, six cases arise:$

$\begin{matrix} Case & Number of ways \\ 1 black, 1 white, 1 red & {}^{3}C_{1} \times {}^{2}C_{1} \times {}^{4}C_{1} = 24 \\ 1 black, 2 white, 0 red & {}^{3}C_{1} \times {}^{2}C_{2} \times {}^{4}C_{0} = 3 \\ 1 black, 0 white, 2 red & {}^{3}C_{1} \times {}^{2}C_{0} \times {}^{4}C_{2} = 18 \\ 2 black, 0 white, 1 red & {}^{3}C_{2} \times {}^{2}C_{0} \times {}^{4}C_{1} = 12 \\ 2 black, 1 white, 0 red & {}^{3}C_{2} \times {}^{2}C_{1} \times {}^{4}C_{0} = 6 \\ 3 black, 0 white, 0 red & {}^{3}C_{3} \times {}^{2}C_{0} \times {}^{4}C_{0} = 1 \end{matrix}$

$∴$ $Total number of ways = 64$

Q 36 :

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points is

540
408
348
296

1

2

3

4

(2)

${}^{30}C_{2} - {}^{8}C_{2} + 1 = \frac{30 \times 29}{2} - \frac{8 \times 7}{2} + 1 = 408$

Q 37 :

Find the number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card.

$\frac{52!}{{(17!)}^{3}}$
$\frac{52!}{{(17!)}^{3} 3!}$
$\frac{51!}{{(17!)}^{3}}$
$\frac{51!}{{(17!)}^{3} 3!}$

1

2

3

4

(2)

52 cards are divided into 3 equal sets (each containing 17 cards) and 1 set (containing only one card), represented as

$\frac{52!}{17! 17! 17! 3! 1!} = \frac{52!}{{(17!)}^{3} 3!}$

Q 38 :

How many numbers greater than 10,00,000 can be formed from 2, 3, 0, 3, 4, 2, 3?

420
360
400
300

1

2

3

4

(2)

$Number of numbers formed = \frac{6 \times 6!}{2! \times 3!} = 360$

Q 39 :

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is

46th
59th
52nd
58th

1

2

3

4

(4)

The number of words in all formed by using the letters of the word SMALL $= \frac{5!}{2!} = 60$

Let's count backwards.

The $59^{th}$ word is SMLAL

$∴$ $58^{th}$ word is SMALL

Q 40 :

The sum $\sum_{r = 1}^{10} (r^{2} + 1) \times (r!)$ is equal to

$11 \times (11!)$
$10 \times (11!)$
$(11!)$
$101 \times (10!)$

1

2

3

4

(2)

We have,

$T_{r} = (r^{2} + 1 + r - r) r! = (r^{2} + r) r! - (r - 1) r!$

$\Rightarrow$ $T_{r} = r (r + 1)! - (r - 1) r!$

$∴$ $T_{1} = 1 \cdot 2! - 0$

$T_{2} = 2 \cdot 3! - 1 \cdot 2!$

$T_{3} = 3 \cdot 4! - 2 \cdot 3!$

.................................................

$T_{10} = 10 \cdot 11! - 9 \cdot 10!$

$∴$ $\sum_{r = 1}^{10} (r^{2} + 1) r! = 10 \cdot 11!$

Q 41 :

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together is

$\frac{7!}{2! 2!}$
$\frac{7!}{2!}$
$\frac{6!}{2!}$
$5! \times 2!$

1

2

3

4

(3)

Required number of ways = $\frac{6!}{2!}$

Q 42 :

The letters of the word COCHIN are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

96
48
183
267

1

2

3

4

(1)

Arranging in alphabetical order → C, C, H, I, N, O

CC..... $\to 4!$

CH..... $\to 4!$

CI..... $\to 4!$

CN..... $\to 4!$

COCHIN $\to 1$

$∴$ No. of words before COCHIN = $4! + 4! + 4! + 4! = 96$

Q 43 :

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is

47th
44th
45th
46th

1

2

3

4

(4)

We have E, E, N, Q, U

According to the English dictionary,

(i) Words starting with E

$E$ ____ ____ ____ ____ $= 4! = 24$

(ii) Words starting with N

$N$ ____ ____ ____ ____ $= \frac{4!}{2!} = 12$

(iii) Words starting with QE

$Q E$ ____ ____ ____ $= 3! = 6$

(iv) Words starting with QN

$Q N$ ____ ____ ____ $= \frac{3!}{2!} = 3$

(v) Word QUEEN = 1

So, according to the English dictionary, the word QUEEN will be at ${(24 + 12 + 6 + 3 + 1)}^{th} = 46^{th}$ position.

Q 44 :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_{1}$ and a particular girl $G_{1}$ never sit adjacent to each other, is

$7!$
$5 \times 6!$
$6 \times 6!$
$5 \times 7!$

1

2

3

4

(2)

$Required number of ways = 7! - 6! \times 2$

$= 7 \times 6! - 6! \times 2 = 5 \times 6!$

Q 45 :

The number of words that can be formed by using all the letters of the word PROBLEM only once is

$5!$
$6!$
$7!$
$8!$

1

2

3

4

(3)

Number of words that can be formed by using all 7 letters of the word PROBLEM only once is $7!$ .

Q 46 :

The possible number of arrangements starting with K of the word KALINGA is

300
330
360
390

1

2

3

4

(3)

Let us fix K at the extreme left position, then we count the arrangements of the remaining 6 letters with A occurring twice.

$∴$ The required number of words starting with K are $\frac{6!}{2!} = 360$ .

Q 47 :

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ________.

(309)

Total words starting from E = $5!$ = 120

Total words starting from H = $5!$ = 120

Total words starting from ME = $4!$ = 24

Total words starting from MH = $4!$ = 24

Total words starting from MOE = $3!$ = 6

Total words starting from MOH = $3!$ = 6

Total words starting from MOR = $3!$ = 6

Total words starting from MOTE = $2!$ = 2

and next word is MOTHER

$∴$ Position of the word 'MOTHER' is

= 120 + 120 + 24 + 24 + 6 + 6 + 6 + 2 + 1

= 240 + 48 + 18 + 3

= 309

Q 48 :

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.

(52)

The units place must be either 0, 4 or 6 for the number to be even.

$\begin{matrix} ↓ & ↓ & ↓ \\ 4 & 5 & 3 \end{matrix} \begin{matrix} \leftarrow No. of ways \end{matrix}$

Number of 3-digit numbers which are even

$= 4 \times 5 \times 3 = 60$

But this includes even numbers with 0 at the hundreds place.

Number of such numbers $= 2 \times 4 = 8$

$∴$ Required number of 3-digit even numbers $= 60 - 8 = 52$

Q 49 :

The digit in the unit's place of the number $1! + 2! + 3! + . . . . . + 99!$ is

3
0
1
7

1

2

3

4

(1)

Unit's place digit of the given number

= Unit's place digit of $(1! + 2! + 3! + 4!) + 0$ (As from $5!$ , unit's place digit = 0)

= Unit's place digit of (1 + 2 + 6 + 24)

= Unit's place digit of 33 = 3

Q 50 :

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ________.

(569)

$a$ $b$ $c$ $d$

Case-I:

$a$ = 2, $b$ = 0, $c$ = 2, $d$ can take values 2, 3, 4, 6, 7 $\to$ so 5 numbers

$a$ = 2, $b$ = 0, $c$ can take values 3, 4, 6, 7, d has 6 choices $\to$ so 24 numbers

Case-II:

$a$ = 2, $b$ can take values 2, 3, 4, 6, 7, $c$ and $d$ have 6 choices each $\to$ so 5 × 6 × 6 = 180 numbers

Case-III:

$a$ = 3, $b$ , $c$ , $d$ each has 6 choices $\to$ so 1 × $6^{3}$ = 216 numbers

Case-IV:

$a$ = 4, $b$ = 0, 2, 3, 4, $c$ and $d$ have 6 choices each $\to$ so 4 × $6^{2}$ = 144 numbers

The total number of 4 digit integers

= 5 + 24 + 180 + 216 + 144

= 569

Q 51 :

The number of arrangements containing all the seven letters of the word ALRIGHT that begins with LG is

720
120
600
540

1

2

3

4

(2)

Required number of ways = $5!$ = 120

Q 52 :

The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

288
1152
625
576

1

2

3

4

(4)

Total letters in the word MACHINE = 7

Number of vowels $= 3, i.e., A, I, E$

Number of ways of arranging vowels $= {}^{4}P_{3} = 4 \times 3 \times 2 = 24$

Number of ways of arranging consonants $= {}^{4}P_{4} = 4! = 24$

$∴$ Required number of ways $= 24 \times 24 = 576$

Q 53 :

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

NRAPGU
NRAGPU
NRAPUG
NRAGUP

1

2

3

4

(1)

Letters in the word NAGPUR = 6

If we fix A _ _ _ _ _

Number of words starting with A = $5!$ = 120

If we fix G _ _ _ _ _

Number of words starting with G = $5!$ = 120

If we fix N A _ _ _ _

Number of words starting with NA = $4!$ = 24

If we fix N G _ _ _ _

Number of words starting with NG = $4!$ = 24

If we fix N P _ _ _ _

Number of words starting with NP = $4!$ = 24

Next word, i.e., 313th word in dictionary will be NRAGPU,

314th word will be NRAGUP,

315th word will be NRAPGU.

Q 54 :

There are 3 different mathematics books and 4 different physics books on a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is

144
120
520
720

1

2

3

4

(4)

Taking all 3 Mathematics books as a single unit and 4 different Physics books as 4 different units, we have 5 units which can be arranged in $5!$ ways.

Also, 3 Mathematics books can be arranged in $3!$ ways.

$∴$ Required number of ways = $5! \times 3! = 120 \times 6 = 720$

Q 55 :

A polygon has 44 diagonals. The number of its sides are

9
8
11
7

1

2

3

4

(3)

$Number of sides = n$

$Number of vertices = n$

$Number of diagonals = {}^{n}C_{2} - n$

$\Rightarrow$ $44 = {}^{n}C_{2} - n \Rightarrow \frac{n (n - 1)}{2} - n = 44$

$\Rightarrow$ $n^{2} - 3 n - 88 = 0$

$∴$ $n = 11$ $(∵ n cannot be negative)$

Q 56 :

The value of $\sum_{r = 1}^{15} r^{2} (\frac{{}^{15}{C_{r}}}{{}^{15}{C_{r - 1}}})$ is equal to

1240
560
1085
680

1

2

3

4

(4)

We have,

$\frac{{}^{15}C_{r}}{{}^{15}C_{r - 1}} = \frac{15!}{r! (15 - r)!} \times \frac{(r - 1)! (15 - r + 1)!}{15!} = \frac{16 - r}{r}$

$∴$ $\sum_{r = 1}^{15} r^{2} (\frac{{}^{15}C_{r}}{{}^{15}C_{r - 1}}) = \sum_{r = 1}^{15} r^{2} (\frac{16 - r}{r}) = \sum_{r = 1}^{15} (16 r - r^{2})$

$= 16 \times \frac{15 \times 16}{2} - \frac{15 \times 16 \times 31}{6} = 680$

Q 57 :

If $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11$ , then $n$ satisfies the equation

$n^{2} - n - 110 = 0$
$n^{2} + 2 n - 80 = 0$
$n^{2} + 3 n - 108 = 0$
$n^{2} + 5 n - 84 = 0$

1

2

3

4

(3)

$We have,$ $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11$

$\Rightarrow$ $\frac{(n + 2) (n + 1) n (n - 1)}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 11$

$\Rightarrow$ $(n + 2) (n + 1) n (n - 1) = 11 \cdot 10 \cdot 9 \cdot 8 \Rightarrow n = 9$

$Now, n = 9 satisfies only n^{2} + 3 n - 108 = 0$

Q 58 :

A password is set with 3 distinct letters from the word LOGARITHMS. How many such passwords can be formed?

90
720
80
72

1

2

3

4

(2)

Required number of passwords formed $= {}^{10}C_{3} \times 3! = 720$

Q 59 :

Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is:

210
25200
2520
302400

1

2

3

4

(2)

3 out of 7 consonants can be chosen in ${}^{7}C_{3}$ ways and 2 out of 4 vowels can be chosen in ${}^{4}C_{2}$ ways.

$∴$ Total no. of words that can be formed $= {}^{7}C_{3} \times {}^{4}C_{2} \times 5!$

$= 25200$

Q 60 :

From 4 men and 6 ladies, a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is:

68
156
60
66

1

2

3

4

(4)

Number of men = 4

Number of ladies = 6

Committee of 5 such that men are in majority can be formed in the following ways.

Case I: Selecting 4 men and 1 lady

Required ways $= {}^{4}C_{4} \times {}^{6}C_{1} = 1 \times 6 = 6$

Case II: Selecting 3 men and 2 ladies

Required ways $= {}^{4}C_{3} \times {}^{6}C_{2} = 4 \times 15 = 60$

$∴$ Total number of ways $= 6 + 60 = 66$

Topic Question Set

Q 31 :

If $\frac{1}{{}^{4}C_{n}} = \frac{1}{{}^{5}C_{n}} + \frac{1}{{}^{6}C_{n}}$ , then $n =$

Q 32 :

${}^{2 n}P_{n}$ is equal to

Q 33 :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

Q 34 :

A man has 6 friends. No. of different ways he can invite 2 or more for a dinner is

Q 35 :

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

Q 36 :

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points is

Q 37 :

Find the number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card.

Q 38 :

How many numbers greater than 10,00,000 can be formed from 2, 3, 0, 3, 4, 2, 3?

Q 39 :

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is

Q 40 :

The sum $\sum_{r = 1}^{10} (r^{2} + 1) \times (r!)$ is equal to

Q 41 :

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together is

Q 42 :

The letters of the word COCHIN are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

Q 43 :

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is

Q 44 :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_{1}$ and a particular girl $G_{1}$ never sit adjacent to each other, is

Q 45 :

The number of words that can be formed by using all the letters of the word PROBLEM only once is

Q 46 :

The possible number of arrangements starting with K of the word KALINGA is

Q 47 :

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ________.

Q 48 :

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.

Q 49 :

The digit in the unit's place of the number $1! + 2! + 3! + . . . . . + 99!$ is

Q 50 :

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ________.

Q 51 :

The number of arrangements containing all the seven letters of the word ALRIGHT that begins with LG is

Q 52 :

The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

Q 53 :

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

Q 54 :

There are 3 different mathematics books and 4 different physics books on a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is

Q 55 :

A polygon has 44 diagonals. The number of its sides are

Q 56 :

The value of $\sum_{r = 1}^{15} r^{2} (\frac{{}^{15}{C_{r}}}{{}^{15}{C_{r - 1}}})$ is equal to

Q 57 :

If $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11$ , then $n$ satisfies the equation

Q 58 :

A password is set with 3 distinct letters from the word LOGARITHMS. How many such passwords can be formed?

Q 59 :

Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is:

Q 60 :

From 4 men and 6 ladies, a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is:

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Topic Question Set

Q 31 : If 1Cn4=1Cn5+1Cn6, then n=

Q 32 : Pn2n is equal to

Q 33 : How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

Q 34 : A man has 6 friends. No. of different ways he can invite 2 or more for a dinner is

Q 35 : A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

Q 36 : Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points is

Q 37 : Find the number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card.

Q 38 : How many numbers greater than 10,00,000 can be formed from 2, 3, 0, 3, 4, 2, 3?

Q 39 : If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is

Q 40 : The sum ∑r=110(r2+1)×(r!) is equal to

Q 41 : The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together is

Q 42 : The letters of the word COCHIN are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

Q 43 : If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is

Q 44 : The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to each other, is

Q 45 : The number of words that can be formed by using all the letters of the word PROBLEM only once is

Q 46 : The possible number of arrangements starting with K of the word KALINGA is

Q 47 : If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ________.

Q 48 : The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.

Q 49 : The digit in the unit's place of the number 1!+2!+3!+.....+99! is

Q 50 : The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ________.

Q 51 : The number of arrangements containing all the seven letters of the word ALRIGHT that begins with LG is

Q 52 : The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

Q 53 : 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

Q 54 : There are 3 different mathematics books and 4 different physics books on a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is

Q 55 : A polygon has 44 diagonals. The number of its sides are

Q 56 : The value of ∑r=115r2(Cr15Cr-115) is equal to

Q 57 : If C6 n+2P2 n-2=11, then n satisfies the equation

Q 58 : A password is set with 3 distinct letters from the word LOGARITHMS. How many such passwords can be formed?

Q 59 : Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is:

Q 60 : From 4 men and 6 ladies, a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is:

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Q 31 :

If $\frac{1}{{}^{4}C_{n}} = \frac{1}{{}^{5}C_{n}} + \frac{1}{{}^{6}C_{n}}$ , then $n =$

Q 32 :

${}^{2 n}P_{n}$ is equal to

Q 33 :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

Q 34 :

A man has 6 friends. No. of different ways he can invite 2 or more for a dinner is

Q 35 :

A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?

Q 36 :

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points is

Q 37 :

Find the number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card.

Q 38 :

How many numbers greater than 10,00,000 can be formed from 2, 3, 0, 3, 4, 2, 3?

Q 39 :

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is

Q 40 :

The sum $\sum_{r = 1}^{10} (r^{2} + 1) \times (r!)$ is equal to

Q 41 :

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together is

Q 42 :

The letters of the word COCHIN are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

Q 43 :

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is

Q 44 :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $B_{1}$ and a particular girl $G_{1}$ never sit adjacent to each other, is

Q 45 :

The number of words that can be formed by using all the letters of the word PROBLEM only once is

Q 46 :

The possible number of arrangements starting with K of the word KALINGA is

Q 47 :

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ________.

Q 48 :

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ________.

Q 49 :

The digit in the unit's place of the number $1! + 2! + 3! + . . . . . + 99!$ is

Q 50 :

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0, 2, 3, 4, 6, 7 is ________.

Q 51 :

The number of arrangements containing all the seven letters of the word ALRIGHT that begins with LG is

Q 52 :

The number of ways in which the letters of the word MACHINE can be arranged such that the vowels may occupy only odd positions is

Q 53 :

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

Q 54 :

There are 3 different mathematics books and 4 different physics books on a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is

Q 55 :

A polygon has 44 diagonals. The number of its sides are

Q 56 :

The value of $\sum_{r = 1}^{15} r^{2} (\frac{{}^{15}{C_{r}}}{{}^{15}{C_{r - 1}}})$ is equal to

Q 57 :

If $\frac{{}^{n + 2}C_{6}}{{}^{n - 2}P_{2}} = 11$ , then $n$ satisfies the equation

Q 58 :

A password is set with 3 distinct letters from the word LOGARITHMS. How many such passwords can be formed?

Q 59 :

Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is:

Q 60 :

From 4 men and 6 ladies, a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is: