Question 1

Find the value of 5 P 2.

Accepted Answer

5 P2 = 5! / (5 - 2)! = 5 x 4 = 20

Question 2

If 50c r = 50c r + 2 find r.

Accepted Answer

By formula, ncx = ncy ⇔ x = y or x + y = n Now, 50cr = 50cr + 2 ⇒ r + r + 2 = 50 or r = r + 2 ⇒ 2r = 48 [∵ r = r + 2 is not possible] ∴ r = 24

Question 3

If nP 3 = 9240, then find the value of n.

Accepted Answer

nP3 = 9240 ⇒ n! / (n-3)! = 9240 ⇒ n(n - 1)(n - 2) = 9240 ⇒ n(n - 1)(n - 2) = 22 x 21 x 20 ∴ n = 22

Question 4

In how many different ways, can the letters of the word 'INHALE' be arranged?

Accepted Answer

The word 'INHALE' has 6 distinct letters. ∴ Number of arrangements = n ! = 6 ! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Question 5

In how many ways, the letters of the word 'ARMOUR' can be arranged?

Accepted Answer

Number of arrangements n ! /( p ! q ! r ! ) Total letters = 6, but R has come twice. So, required number of arrangements = 6 ! / 2 ! = (6 x 5 x 4 x 3 x 2 !) / 2 ! = 360

Question 6

In how many ways, the letters of the word 'STRESS' can be arranged?

Accepted Answer

Required number of arrangements = 6 ! / 3 ! [∵ S has come thrice ] = [ 6 x 5 x 4 x 3! ] /3 ! = 120

Question 7

In how many ways, the letters of the word 'BANKING' can be arranged?

Accepted Answer

Total letters = 7, but N has come twice. So, required number of arrangements = 7 ! / 2 ! = [7 x 6 x 5 x 4 x 3 x 2 ! ] / 2 ! = 2520

Question 8

In how many different ways, the letters of word 'FINANCE' can be arranged?

Accepted Answer

Total number of letters = 7, but N has come twice. So, required number of arrangements = 7 ! / 2 ! = [7 x 6 x 5 x 4 x 3 x 2! ] / 2! = 2520

Question 9

In how many different ways, can the letters of the word 'VENTURE' be arranged?

Accepted Answer

The required different ways = 7 ! / 2 ! = 7 x 6 x 5 x 4 x 3 x 2!/2! = 2520

Question 10

In a meeting between two countries, each country has 12 delegates. All the delegates of one country shake hands with all delegates of the other country. Find the number of handshakes possible?

Accepted Answer

Total number of handshakes = 12 x 12 = 144

Question 11

How many straight lines can be drawn from 15 non-collinear points?

Accepted Answer

Required number of lines = nC2 = 15C2 = (15 x 14)/2 = 105

Question 12

A library has 'a' copies of one book, 'b' copies of each of two book, 'c' copies of each of three books and single copy of 'd' book. The number of ways in which these books can be distribute is?

Accepted Answer

Total number of books = a + 2b + 3c + d . Since there are 'b' copies of each of two books, 'c' copies of each of three books and single copy of 'd' book. Therefore, the total number of arrangements is = (a + 2b + 3c + d )! / {a! (b!)2 (c!)3}

Question 13

The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is?

Accepted Answer

Required no. of triangles = 12C3 - 7C3 = 185

Question 14

How many words of 4 consonants and 3 vowels can be made from 12 consonants and 4 vowels. If all the letters are different?

Accepted Answer

Required no. of words = 12P4 x 4P3 = 12 x 11 x 10 x 9 x 4 x 3 x 2 = 120(100 - 1) x 24 = 288000 - 2880 = 285120

Question 15

12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is?

Accepted Answer

12 persons can be seated around a round table in 11! ways. The total number of ways in which 2 particular persons sit side by side = 10! x 2!. Hence, the required number of arrangements = 11! - 10! x 2! = 9 x (10!)

Question 16

The total number of selections of fruit which can be made from 3 bananas, 4 apples and 2 oranges is?

Accepted Answer

Required number of ways = (2 + 1) (3 + 1) (4 + 1) - 1 = 59

Question 17

The total number of arrangements of the letters in the expression a 3b 2c 4 when written at full length is?

Accepted Answer

We have 9 letters 3 a's, 2 b's and 4 c's. These 9 letters can be arranged in 9! / (3! 2! 4!) = 1260 ways.

Question 18

If nC r - 1 = 36, nC r = 84 and nC r + 1 = 126 then?

Accepted Answer

nCr - 1 = 36, nCr = 84 and nCr + 1 = 126 ⇒ nCr / nCr - 1 = 84/36 ⇒ 3n - 10r + 3 = 0 ....(i) nCr + 1 / nCr = 126/84 ⇒ 2n - 5r - 3 = 0 .............(ii) From (i) and (ii) r = 3, n = 9

Question 19

If 35C n + 7 = 35C 4n - 2, then all the values of n are given by?

Accepted Answer

35Cn + 7 = 35C4n - 2 We know that if nCx = nCy then, Either x = y or x + y = n Case 1: x + y = n ⇒ n + 7 + 4n - 2 = 35 ⇒ 5n + 5 = 35 ⇒ 5n = 30 ∴ n = 6 Case 2: x = y ⇒ n + 7 = 4n - 2 ⇒ 4n - n = 2 + 7 ⇒ 3n = 9 ∴ n = 3

Question 20

If 2n + 1P n - 1 : 2n - 1P n = 3 : 5, then n is equal to?

Accepted Answer

We have If 2n + 1Pn - 1 / 2n - 1Pn = 3/5 ⇒ 5 x 2n + 1Pn - 1 = 3 x 2n - 1Pn ⇒ {5 x (2n + 1)!} / {(n + 2 )!} = 3 x {(2n - 1)!} / {(n - 1)!} ⇒ 10(2n + 1) = 3(n + 2) (n + 1) ⇒ 3n2 - 11n - 4 = 0 ⇒(3n + 1 ) (n - 4) = 0 ∴ n = 4

Permutation and Combination Questions