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Q.

Set $A$ has $m$ elements and set $B$ has $n$ elements. If the total number of subsets of A is $112$ more than the total number of subsets of $B$ , then the value of $m n$ is .…..

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answer is 28.

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Detailed Solution

$o (A) = m \Rightarrow o [S (A)] = 2^{m} and o (B) = n \Rightarrow o [S (B)] = 2^{n}$

Given $2^{m} = 2^{n} + 112 \Rightarrow 2^{m} - 2^{n} = 112 \Rightarrow 2^{n} (2^{m - n} - 1) = 7 \times 2^{4} \Rightarrow n = 4, m - n = 3$

$\Rightarrow m = 7, n = 4 . ∴ m n = (7) (4) = 28$ .

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Set Ahas m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of mn is .…..