Two finite sets have \(m\) and \(n\) elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The value of \(m\) and \(n\) respectively are:
(a) \(7,6\)
(b) \(5,1\)
(c) \(6,3\)
(d) \(8,7\)
Solution
Number of subsets of a set having \(m\) elements:
\[ 2^m \]
Number of subsets of a set having \(n\) elements:
\[ 2^n \]
Given,
\[ 2^m-2^n=56 \]
Checking the options:
For \(m=6,\ n=3\),
\[ 2^6-2^3 \]
\[ =64-8 \]
\[ =56 \]
Hence,
\[ m=6,\qquad n=3 \]
Answer
\[ \boxed{6,3} \]
Correct option: (c)