Let A = 1,2,3,4,5,6,7.

Question

Let $A = \{1,2,3,4,5,6,7\}$. Define $B = \{T\subseteq A : 	ext{either } 1
otin T 	ext{ or } 2\in T\}$ and $C = \{T\subseteq A : 	ext{the sum of all the elements of } T 	ext{ is a prime number}\}$. Then the number of elements in the set $B\cup C$ is \_\_\_\_.

Answer: 107

Accepted Answer

$(B\cup C)' = B'\cap C'$. Here $B' = \{T : 1\in T \text{ and } 2\notin T\}$, and $C' = \{T : \text{sum of elements is not prime}\}$.

We need subsets $T'$ of $\{3,4,5,6,7\}$ (since $1$ is forced in, $2$ forced out) such that $1+\text{sum}(T')$ is not prime. Counting by size of $T'$:

, size of $T'$, valid count |
---|---|
5|1|
4|3|
3|6|
2|7|
1|3|
0|1|

$n(B'\cap C') = 1+3+6+7+3+1 = 21$

$$n(B\cup C) = 2^7 - 21 = 128-21 = 107$$

Solutions · 1