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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 set. The values of m and n are, respectively ______.
Options
4, 7
7, 4
4, 4
7, 7
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Solution
Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively 7, 4.
Explanation:
Number of subsets of a given set having m element = 2m and the number of subsets of set containing n elements = 2n
As per the given condition, we have
2m – 2n = 112
⇒ 2n (2m – n – 1) = 112
⇒ 2n .(2m – n – 1) = 24 .7
⇒ 2n = 24 and 2m – n – 1 = 7
⇒ n = 4 and 2m – n = 1 + 7 = 8 = 23
⇒ n = 4 and m – n = 3
⇒ m – 4 = 3
⇒ m = 7
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