Question

Two finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. The values of m and n are?
${\text{A}}{\text{. 7,6}} \\\ {\text{B}}{\text{. 6,7}} \\\ {\text{C}}{\text{. 6,4}} \\\ {\text{D}}{\text{. 7,4}} \\\$

Answer 1

Hint- No. of subsets of a finite set of $x$elements is ${2^x}$.We have to find the no of subsets of both the finite sets of given no of elements and form an equation that satisfies the given statement.

Complete step-by-step solution -
No. of subsets for first set $= {2^m}$
No. of subsets for second set $= {2^n}$
Now, according to question,
No. of subsets of first set is 48 more than that of second set

$${2^m} = 48 + {2^n} \\\ {2^m} - {2^n} = 48 \\\ {\text{Now factorize 48 and take }}{{\text{2}}^n}{\text{ common from left side}} \\\ {2^n}(\dfrac{{{2^m}}}{{{2^n}}} - 1) = 16 \times 3 \\\ {2^n}({2^{m - n}} - 1) = {2^4} \times (4 - 1) \\\ {2^n}({2^{m - n}} - 1) = {2^4} \times ({2^2} - 1) \\\$$

Here we have factorized 48 in such a way that we can get the value of m and n by comparing LHS and RHS.
Comparing both sides, we get
$n = 4 \\\ {\text{and }}m - n = 2 \\\ m - 4 = 2 \\\ m = 4 + 2 \\\ m = 6 \\\$
Therefore, the correct option is C.

Note- For solving such a question try to consider the number of elements in some unknown variable and with the help of the problem statement try to bring out some algebraic equation. The formula for number of subsets for any given set must be remembered.