Question

Two finite sets have m and a elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and a respectively are:

• A. 45479
• B. 45413
• C. 45446
• D. 8,7

Answer 1

The Correct Option is C

The number of subsets of a set with $n$ elements is $2^n$.
Therefore, we have: $2^m - 2^a = 56$ We can factor the left-hand side to get: $2^a(2^{m-a} - 1) = 56$ 
Since $2^a$ and $2^{m-a} - 1$ are integers, we can factor 56 into its prime factors: $56 = 2^3 \cdot 7$ 
We can then try different values of $a$ and $m-a$ to see which ones satisfy the equation.
If $a = 1$, then $2^{m-a} - 1 = 28$, which has no integer solution for $m-a$.
If $a = 2$, then $2^{m-a} - 1 = 14$, which has no integer solution for $m-a$.
If $a = 3$, then $2^{m-a} - 1 = 7$, which has the solution $m-a = 3$.
Therefore, $m = 6$ and $a = 3$.