Question

A set contains (2n + 1) elements. If the number of subsets of this set which contain at most n elements is 4096, then the value of n is

• A. 6
• B. 15
• C. 21
• D. None of these

Answer 1

Answer: (A)

The number of subsets of the set which contain at most n elements is
${}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+....{+}^{2n+1}{C}_n=K\left(say\right)$
We have
$2K=2\left({}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+....{+}^{2n+1}{C}_n\right)$
$=\left({}^{2n+1}{C}_0{+}^{2n+1}{C}_{2n+1}\right)+\left({}^{2n+1}{C}_1{+}^{2n+1}{C}_{2n}\right)$
$+...+\left({}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}\right)\left({\because }^n{C}_r{=}^n{C}_{n-r}\right)$
${=}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+....{+}^{2n+1}{C}_{2n+1}$
$=2^{2n+1}\Rightarrow K=2^{2n}$