Question

A student is allowed to select at most n books from a collection of $(2n+1)$ books. If the total number of ways in which he can select at least one book is $225$, then the value of $n$ is equal to

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The correct option is D

$3$

The explanation for the correct answer:

Step 1: Finding the number of ways of selection:

A student is allowed to select most $n$ books out of $(2n+1)$ books.

This selection is given by the combination concepts,

${}^{2n+1}C_1+{}^{2n+1}C_2+{}^{2n+1}C_3+........+{}^{2n+1}C_n=63$ …(1)

we know that , ${}^{n}C_r={}^{n}C_{n-r}$. and

${}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+........+{}^{2n+1}C_{2n+1}=2^{2n+1}$.

Step 2: Finding the value of $n$:

Thus the equation becomes,

$$\begin{gathered} \Rightarrow 2({}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+........+{}^{2n+1}C_n)=2^{2n+1} \\ \Rightarrow {}^{2n+1}C_0+{}^{2n+1}C_1+{}^{2n+1}C_2+........+{}^{2n+1}C_n=2^{2n} \\ \Rightarrow {}^{2n+1}C_1+{}^{2n+1}C_2+{}^{2n+1}C_3+........+{}^{2n+1}C_n=2^{2n}-1\left[\because {}^{2n+1}C_0=1\right] \end{gathered}$$

On comparing equation 1 with this,

$$\begin{gathered} \Rightarrow 2^{2n}-1=63 \\ \Rightarrow 2^{2n}=64 \\ \Rightarrow 2^{2n}=2^6 \\ \Rightarrow 2n=6 \\ \Rightarrow n=\mathbf{3} \end{gathered}$$

Hence, option (D) is the correct answer.