Question

A student is allowed to select almost n books from a collection of 2n + 1 books. If the total no. of w

• A. 6
• B. 3
• C. 4
• D. 5

Answer 1

¼b½

Sol.

$\mathbf{2n + 1}\mathbf{C}_{\mathbf{1}}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{2}}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{3}}\mathbf{+ ........}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{n}}\mathbf{= 63}\mathbf{2n + 1}\mathbf{C}_{\mathbf{0}}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{1}}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{2}}\mathbf{+ .........}\mathbf{+}^{\mathbf{2n + 1}}\mathbf{C}_{\mathbf{2n + 1}}\mathbf{=}\mathbf{2}^{\mathbf{2n + 1}}$

No. of terms is even. Sum of first half = sum of second half.

$2n + 1C_{0} + 63 = 2^{2n}$.

63 + 1 = 2²ⁿ

2²ⁿ = 64 = 2⁶

n = 3