A student is allowed to choose atmost $n$ books from a collection of $2 n+1$ books. If the total number of ways in which he can select atleast one book is 255 , then the value of $n$ is

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Aicharya · Accepted Answer

According to given information, $\begin{aligned} & { }^{2 n+1} C_1+{ }^{2 n+1} C_2+{ }^{2 n+1} C_3+\ldots \ldots+{ }^{2 n+1} C_n=x \text{ Let}\ & \because^{2 n+1} C_0+{ }^{2 n+1} C_1+{ }^{2 n+1} C_2+\ldots+{ }^{2 n+1} C_n+\ldots \ & +{ }^{2 n+1} C_{2 n+1}=2^{2 n+1} \ & \Rightarrow 2 x+{ }^{2 n+1} C_0+{ }^{2 n+1} C_{2 n+1}=2^{2 n+1} \ & {\left[\because{ }^n C_{n-r}={ }^n C_r\right]} \ & \Rightarrow \quad 2 x+1+1=2^{2 n+1} \ & {\left[\because{ }^n C_0={ }^n C_n=1\right]} \ & \Rightarrow \quad x=2^{2 n}-1=255 \quad \text{(given)}\ & \Rightarrow \quad 2^{2 n}=256=2^8 \ & \Rightarrow \quad n=4 \ \end{aligned}$ Hence, option (1) is correct.

A student is allowed to choose atmost \(n\) books from a collection of \(2 n+1\) books. If the total number…

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