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

Correct Answer: 107

$A:\{1,2,3,4,5,6,7\}$
Number of elements in set B
$=n(1\notin T)+n(2\in T)-n[(1\notin T)\cap (2\in T)]$
$=2^6+2^6-2^5=96$
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 $C=42$
$\Rightarrow n(B\cup C)=n(B)+n(C)-n(B\cap C)$
$=96+42-31=107$

The correct answer is 107.