Question

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

Answer 1

A:{1,2,3,4,5,6,7}
Number of elements in set B
=n(1∈/T)+n(2∈T)−n[(1∈/T)∩(2∈T)]
=26+26−25=96
Number of elements in set C
={{2},{3},{5},{7},{1,2},{1,4},{1,6},
{2,3},{2,5},{3,4},{4,7},{5,6},{6,7}
{1,2,4},{1,3,7},{1,4,6},{1,5,7},{2,3,
6},{2,4,5},{2,4,7},{2,5,6},{3,4,6},
{4,6,7},{1,2,4,6},{2,4,6,7},{2,4,6,
5},{3,5,7,4},{1,3,5,4},{1,5,7,4},{1,
2,3,5},{1,2,3,7},{1,3,6,7},{1,5,6,7},
{2,3,5,7},{1,5,7,2,4},{3,5,7,2,6},{1,
3,7,2,4},{1,4,5,6,7},
{1,3,4,5,6},{1,2,3,6,7},{1,2,3,5,6},
{1,2,3,4,6,7}
Number of elementrrs in C=42
⇒n(B∪C)=n(B)+n(C)−n(B∩C)
=96+42−31=107

The correct answer is 107.