Question

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

Answer 1

No. of subsets of Set A of m elements = 2^m

Again No. of subsets of Set B of n elements = 2ⁿ

Set A have 56 more subsets than Set B.

It means 2^m−2ⁿ = 56

Then it is clear m > n.

Let m = n + k,

Now 2^m−2ⁿ = 56

or 2^n+k - 2ⁿ = 56

or 2ⁿ(2^k−1) =56

or 2ⁿ(2^k−1) = 8×7 =2³(2³−1)

Thus we see n=3, k=3

m = n+k= 3+3 = 6


Ans. n= 3, m = 6