Let A = {1, 2, 3, 4, 5, 6, 7}. Define B = $B=\left\{T\subseteq A:either1\notin Tor2\in T\right\}$ and $C=\left\{T\subseteq A:TthesumofalltheelementsofTisaprimenumber\right\}.$ Then the number of elements in the set $B\cup C$ is

<p><span contenteditable="false"> <math> <mrow> <mo>? </mo> <mrow> <mo> (</mo> <mrow> <mi>B</mi> <mo>∪</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>'</mo> <mo>=</mo> <mi>B</mi> <mo>'</mo> <mo>∪</mo> <mi>C</mi> <mo>'</mo> </mrow> </math> </span></p><p>B’ is a set containing sub sets of A containing element 1 and not containing 2.</p><p>And C is a set containing subsets of A whose sum of elements is not prime.</p><p>&nbsp;so, we need to calculate number of subsets of</p><p> {3, 4, 5, 6, 7} whose sum of elements plus 1 is composite.</p><p>Number of such 5 elements subset = 1</p><p>Number of such 4 elements subset = 3 (except selecting 3 or 7)</p><p>Number of such 3 elements subset = 6 (except selecting {3, 4, 5}, {3, 6, 7}, {4, 5, 7} or {5, 6, 7})</p><p>Number of such 2 elements subset = 7 (except selecting {3, 7}, {4, 6}, {5, 7})</p><p>Number of such 1 elements subset = 3 (except selecting {3, 7}, (4, 6}, {5, 7})</p><p>Number of such 0 elements subset = 1</p><p><span contenteditable="false"> <math> <mrow> <mi>n</mi> <mrow> <mo> (</mo> <mrow> <mi>B</mi> <mo>∩</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mn>1</mn> <mo>⇒</mo> <mi>n</mi> <mrow> <mo> (</mo> <mrow> <mi>B</mi> <mo>∪</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mn>2</mn> </mrow> <mrow> <mn>7</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mn>0</mn> <mn>7</mn> </mrow> </math> </span></p>

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