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(C) The number of subsets of a set with $k$ elements is $2^k$.Given that the number of subsets of $A$ is $112$ more than the number of subsets of $B$,

Vedclass · Accepted Answer

$28$

Vedclass · Answer

(C) The number of subsets of a set with $k$ elements is $2^k$.Given that the number of subsets of $A$ is $112$ more than the number of subsets of $B$,we have the equation: $2^m - 2^n = 112$.We can rewrite this as $2^n(2^{m-n} - 1) = 112$.Since $112 = 16 	imes 7 = 2^4 	imes (2^3 - 1)$,we compare the terms:$2^n = 2^4 \implies n = 4$.$2^{m-n} - 1 = 2^3 - 1 \implies m - n = 3$.Substituting $n = 4$,we get $m - 4 = 3$,so $m = 7$.Therefore,$m 	imes n = 7 	imes 4 = 28$.

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 $m \times n$ is

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