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Question
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
Options
(a) 4, 7
(b) 7, 4
(c) 4, 4
(c) 4, 4
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Solution
We know that if a set X contains k elements, then the number of subsets of X are 2k.
It is given that the number of subsets of a set containing m elements is 112 more than the number of subsets of set containing n elements.
\[\therefore 2^m - 2^n = 112\]
\[ \Rightarrow 2^n \left( 2^{m - n} - 1 \right) = 2 \times 2 \times 2 \times 2 \times 7\]
\[ \Rightarrow 2^n \left( 2^{m - n} - 1 \right) = 2^4 \left( 2^3 - 1 \right)\]
\[ \Rightarrow n = 4 \text{ and } m - n = 3\]
\[ \therefore m - 4 = 3\]
\[ \Rightarrow m = 7\]
Thus, the values of m and n are 7 and 4, respectively.
Hence, the correct answer is option (b).
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