Question

Two finite sets have \(m\) and \(n\) elements. Then total number of subsets of the first set is \(56\) more than that of the total number of subsets of the second. The value of \(m\) and \(n\) are

• A. \(7,6\)
• B. \(6,3\)
• C. \(5,1\)
• D. \(8,7\)

Hint:

A set with \(k\) elements has \(2^k\) subsets. Convert subset problems into powers of 2.

Answer 1

The Correct Option is B

Concept:
If a set contains \(k\) elements, then the total number of subsets of that set is: \[ 2^k \] Here, the first set has \(m\) elements and the second set has \(n\) elements. So: \[ \text{Subsets of first set}=2^m \] \[ \text{Subsets of second set}=2^n \]

Step 1: Convert the statement into equation.
Given: Total number of subsets of the first set is \(56\) more than that of the second set. So: \[ 2^m=2^n+56 \] Therefore: \[ 2^m-2^n=56 \]

Step 2: Check option (A).
For \((m,n)=(7,6)\): \[ 2^7-2^6=128-64=64 \] This is not \(56\). So option (A) is incorrect.

Step 3: Check option (B).
For \((m,n)=(6,3)\): \[ 2^6-2^3=64-8=56 \] This satisfies the condition. So option (B) is correct.

Step 4: Check remaining options quickly.
For \((5,1)\): \[ 2^5-2^1=32-2=30 \] For \((8,7)\): \[ 2^8-2^7=256-128=128 \] Neither gives \(56\). Hence, the correct answer is: \[ \boxed{(B)\ 6,3} \]