If for two sets A and B, n(A) = m and n(B) = n and (Number of subsets of A – Number of subsets of B) = 56, then find value of (2m + n).

The correct answer is 15.

Let us break down the solution step-by-step:

1. Determine the number of subsets for each set:
For any finite set $S$ containing $k$ elements, the total number of subsets is given by $2^k$.
Here, set $A$ has $m$ elements, so the number of subsets of $A$ is:
$n\left(P\right(A\left)\right)=2^m$
Similarly, set $B$ has $n$ elements, so the number of subsets of $B$ is:
$n\left(P\right(B\left)\right)=2^n$

2. Set up the equation from the given condition:
We are given that the difference between the number of subsets of $A$ and the number of subsets of $B$ is 56. Thus, we can write:
$2^m-2^n=56$
Since 56 is positive, we must have $m>n$. We can factor out $2^n$ from the left-hand side:
$2^n(2^{m-n}-1)=56$

3. Express 56 as a product of a power of 2 and an odd number:
Let us find the prime factorization of 56:
$56=8\times 7=2^3\times 7$
Substituting this back into the equation:
$2^n(2^{m-n}-1)=2^3\left(7\right)$

4. Solve for m and n by comparing factors:
Comparing the even power of 2 components:
$2^n=2^3\Rightarrow n=3$
Comparing the odd factor components:
$2^{m-n}-1=7$
Adding 1 to both sides gives:
$2^{m-n}=8$
Since $8=2^3$, we have:
$m-n=3$
Substitute the value of $n=3$:
$m-3=3\Rightarrow m=6$

5. Calculate the required value of (2m + n):
Substitute $m=6$ and $n=3$ into the expression:
$2m+n=2\left(6\right)+3=12+3=15$