First slide

Combinations

Question

A student is allowed to select atmost n books from a collection of (2n + 1) books. If the total number of ways in which he can select books is 63, then n =

Moderate

A

4
B

3
C

7
D

8

Solution

The number of ways in which at most n books can be selected out of a collection of (2n + 1) books is
$\begin{array}{l} ^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots^{2 n + 1} C_{n} \\ \Rightarrow \frac{1}{2} [(^{2 n + 1} C_{1} +^{2 n + 1} C_{1}) + (^{2 n + 1} C_{2} +^{2 n + 1} C_{2}) + \dots \dots + (^{2 n + 1} C_{n} +^{2 n + 1} C_{n})] = 63 \\ \Rightarrow (^{2 n + 1} C_{1} +^{2 n + 1} C_{2 n}) + (^{2 n + 1} C_{2} +^{2 n + 1} C_{2 n - 1}) + \dots \dots + (^{2 n + 1} C_{n} +^{2 n + 1} C_{n + 1}) = 126 \\ \begin{array}{l} \Rightarrow^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{2 n} = 126 \\ \Rightarrow 2^{2 n + 1} - 2 = 126 \\ \Rightarrow 2^{2 n + 1} = 128 = 2^{7} \\ \Rightarrow 2 n + 1 = 7 \\ \Rightarrow n = 3 \end{array} \end{array}$

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