Q60 / 160 · TS EAMCET 2023 (12 May Shift 2) · Mathematics
A student is allowed to select at most $\mathrm{n}$ books from a collection of $(2 n+1)$ books. If the total number of ways in which ne can select at least one book is 255 , then the value of $\mathrm{n}$ is
- Correct answerA$4$
- B$5$
- C$6$
- D$7$
Solution
$$
\begin{aligned}
& \quad{ }^{2 n+1} C_1+{ }^{2 n+1} C_2+\ldots . .+{ }^{2 n+1} C_n=255 (i)\\
& \because{ }^n C_r={ }^n C_{n-r} \\
& { }^{2 n+1} C_{2 n}+{ }^{2 n+1} C_{2 n-1}+\ldots .+{ }^{2 n+1} C_{n+1}=255 (ii)
\end{aligned}
$$
Add (i) and (ii),
$$
\begin{aligned}
& \left({ }^{2 n+1} C_1+{ }^{2 n+1} C_2+\ldots .+{ }^{2 n+1} C_{2 n}\right)=510 \\
& \Rightarrow{ }^{2 n+1} C_0+{ }^{2 n+1} C_1+\ldots .+{ }^{2 n+1} C_{2 n}+{ }^{2 n+1} C_{2 n+1}=512 \\
& \Rightarrow 2^{2 n+1}=512=2^9 \\
& \therefore \quad 2 n+1=9 \\
& \therefore \quad n=4 .
\end{aligned}
$$
Hence, the correct answer is option: 'A'.