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  4. Two finite sets have m and n elements. The number of subsets of the first set is 224 more than that of the second set. The values of m and n are, respectively.

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Q. Two finite sets have $m$ and $n$ elements. The number of subsets of the first set is $224$ more than that of the second set. The values of $m$ and $n$ are, respectively.

Solution:

$2^{m}-2^{n}=224$
$2^{n}\left(2^{m-n}-1\right)=2^{5} \times 7$
$2^{n}\left(2^{m-n}-1\right)=2^{5}\left(2^{3}-1\right)$
$n=5 \,\,\,m-n=3 \Rightarrow m=8$