Two finite sets have m and n elements.

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.

$\therefore 2^{m}-2^{n}=112$

$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=2 \times 2 \times 2 \times 2 \times 7$

$\Rightarrow 2^{n}\left(2^{m-n}-1\right)=2^{4}\left(2^{3}-1\right)$

$\Rightarrow n=4$ 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).