The total no. of selections of at most n things from 2n+1 different things is found to be 63. find the value of n.

The number of ways of selecting at least one thing are


$2n+1_{C_1}+2n+1_{C_2}+...... 2n+1_{C_n} = 63 = S \left(\mathrm{let}\right)$

Again we know that,

$2n+1_{C_0}+2n+1_{C_1}+...... 2n+1_{C_n}+......+2n+1_{C_{2n+1}} = 2^{2n+1}$

Now,

$$\begin{gathered} 2n+1_{C_0} = 2n+1_{C_{2n+1}} = 1 (\mathrm{because} \mathrm{first} \mathrm{term} = \mathrm{last} \mathrm{term}) \\ 2n+1_{C_1} = 2n+1_{C_{2n}} \mathrm{etc}.... (\mathrm{second} \mathrm{term} = \mathrm{second} \mathrm{last} \mathrm{term}) \end{gathered}$$

Remaining terms will lead to 2S.

So we have,

$$\begin{gathered} 1+1+2S = 2^{2n+1} \\ \Rightarrow 2+2\times 63 = 2^{2n+1} \{\mathrm{since} S = 63 \mathrm{from} \mathrm{above}\} \\ \Rightarrow 128 = 2^{2n+1} \\ \Rightarrow 2^7 = 2^{2n+1} \\ \Rightarrow 2^{2\times 3+1} = 2^{2n+1} \\ \mathrm{Comparing} \mathrm{both} \mathrm{sides} \mathrm{of} \mathrm{equation} \mathrm{we} \mathrm{get}; \\ n = 3 \end{gathered}$$

Therefore the value of n is 3.