Question

The number of elements in sets X and Y are $p$ and $q$ respectively. The total number of subsets of X is 112 more than that of Y. What is the value of $(2p - 3q)$?

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

Use powers of 2 to reverse-engineer values of $p$ and $q$ from the subset count.

Answer 1

The Correct Option is A

Number of subsets of a set with $n$ elements = $2^n$
Given: $2^p = 2^q + 112$
Try small values: $2^7 = 128$, $2^6 = 64$ → $128 - 64 = 64$ (too low)
$2^8 = 256$, $2^7 = 128$ → $256 - 128 = 128$
$2^7 = 128$, $2^6 = 64$ → $128 - 64 = 64$
Eventually, we find $2^p = 128$, $2^q = 16$ → $p = 7$, $q = 4$
$\Rightarrow 2p - 3q = 2(7) - 3(4) = 14 - 12 = 2$