Two finite sets have \(m\) and \(n\) elements. The number of subsets of the first set is 112 more than that of the second. The values of \(m\) and \(n\) are respectively
(a) \(4,7\)
(b) \(7,4\)
(c) \(4,4\)
(d) \(7,7\)
Solution
Number of subsets of a set having \(m\) elements:
\[ 2^m \]
Number of subsets of a set having \(n\) elements:
\[ 2^n \]
Given,
\[ 2^m-2^n=112 \]
Checking the options:
For \(m=7,\ n=4\),
\[ 2^7-2^4 \]
\[ =128-16 \]
\[ =112 \]
Hence,
\[ m=7,\qquad n=4 \]
Answer
\[ \boxed{7,4} \]
Correct option: (b)