Discussion
Let A ={a,b,c,d,e,f} Then the number of subsets of A with an odd number of elements is
Answer: E
There are 32 subsets with an odd number of elements.
A set of \(n\) elements has \(2^{n}\) subsets, split equally between odd and even sizes, because \(\sum (-1)^{k}\,{}^{n}C_{k} = 0\). With \(n = 6\):
\[ \begin{aligned} {}^{6}C_{1} + {}^{6}C_{3} + {}^{6}C_{5} &= 6 + 20 + 6 \\ &= 32 \\ &= \frac{2^{6}}{2} \end{aligned} \]
The other options are partial counts: 12 is \({}^{6}C_{1} + {}^{6}C_{5}\), and 8, 16 and 24 do not match the sum.
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