Question

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively,

A. 4, 7
B. 7, 4
C. 4, 4
D. 7, 7

Answer 1

B. 7, 4

Given: Two finite sets have m and n elements.

To find: value of m and n

Formula used:

The number of subsets of a set containing x elements is given by 2^x

According to question:

2^m – 2ⁿ = 112

⇒ 2ⁿ (2^m-n – 1) = 16 × 7

⇒ 2ⁿ (2^m-n – 1) = 2⁴ × 7

On comparing:

2ⁿ = 2⁴ and 2^m-n – 1 = 7

⇒ n = 4 and 2^m-n = 8

⇒ 2^m-n = 2³

⇒ m – n = 3

⇒ m – 4 = 3

⇒ m = 7

Hence, value of m and n is 7 and 4 respectively