1. Consider all 2^{n} – 1 non-empty subsets of the set{1, 2, 3, .... n }
product of each of its elements. Sum of all these
products is

a) (n + 1)!

b) (n + 1)! + 1

c) (n + 1)! – 1

d) n! – 2

Explanation: Consider the product (1 + 1) (1 + 2) .... (1 + n)

2. Consider all \[2^{n}-1\] non-empty subsets of the set
{1, 2, .... , n} . For every such subset we find the
product of reciprocals of each of its elements. Sum
of all these products is

a) n

b) n + 1

c) n!

d) n!-1

Explanation:

3. The number of ways in which we can arrange 4
letters of the word MATHEMATICS is given by

a) 136

b) 2454

c) 1680

d) 192

Explanation:

4. The number of ways in which we can distribute mn
students equally among m sections is given by

a) \[\frac{\left(mn\right)!}{n!}\]

b) \[\frac{\left(mn\right)!}{\left(n!\right)^{m}}\]

c) \[\frac{\left(mn\right)!}{m!n!}\]

d) \[\left(mn\right)^{m}\]

Explanation:

5. If a polygon has 90 diagonals, the number of its
sides is given by

a) 12

b) 11

c) 10

d) 15

Explanation:

6. Out of 10 white, 8 black and 6 red balls, the number
of ways in which one or more balls can be selected
is given by

a) 681

b) 691

c) 679

d) 692

Explanation: (10 + 1) (8 + 1) (6 + 1) - 1

7. A is a set containing n elements. A subset P of A is
chosen. The set A is reconstructed by replacing the
elements of P. A subset Q of A is again chosen. The
number of ways of choosing P and Q so that \[ P\cap Q \]
contains exactly two elements is

a) \[9\times ^{n}C_{2}\]

b) \[3^{n}- ^{n}C_{2}\]

c) \[2\times ^{n}C_{n}\]

d) \[ ^{n}C_{2}.3^{n-2}\]

Explanation:

8. Ten different letters of an alphabet are given. Words with five letters are formed from these given letters .
The number of words which have at least one of
their letters repeated is

a) 69760

b) 30240

c) 99748

d) 60480

Explanation:

9. The number of ways in which we can select four
numbers from 1 to 30 so as to exclude every
selection of four consecutive numbers is

a) 27378

b) 27405

c) 27397

d) 19050

Explanation:

10. If \[ ^{n}C_{4}\] , \[ ^{n}C_{5}\] and \[ ^{n}C_{6}\] are in A.P., the value of n can be

a) 14

b) 11

c) 9

d) 5

Explanation: