The coefficient of xn in the polynomial (. x + 2 n + 1 C0 .) (. x + 2 n + 1 C1 .) (. x + 2 n + 1 C2 .) ..... (. x + 2 n + 1 Cn .) is

Question

The coefficient of xn in the polynomial (. x + 2 n + 1 C0 .) (. x + 2 n + 1 C1 .) (. x + 2 n + 1 C2 .) ..... (. x + 2 n + 1 Cn .) is (A) 2n + 1 (B) 22 n + 1 - 1 (C) 22 n - 1 (D) 22 n

Tardigrade Admin · Accepted Answer

Given Expression is, (. x + 2 n + 1 C0 .) (. x + 2 n + 1 C1 .) (. x + 2 n + 1 C2 .) ..... (. x + 2 n + 1 Cn .) If P is coefficient of xn then, P = 2 n + 1 C0 + 2 n + 1 C1 + 2 n + 1 C2 + ..... + 2 n + 1 Cn -------(1) ⇒ P = 2 n + 1 C2 n + 1 + 2 n + 1 C2 n + 2 n + 1 C2 n - 1 + ..... + 2 n + 1 Cn + 1 ------(2) (. ∵ n Cr =n Cn - r .) adding (1) and (2) 2 P = (. 2 n + 1 C0 +2 n + 1 C1 + ....... +2 n + 1 C2 n + 1 .) 2 P = 22 n + 1 ∴ P=22 n

Q. The coefficient of $x^{n}$ in the polynomial $(x +^{2 n + 1} C_{0}) (x +^{2 n + 1} C_{1}) (x +^{2 n + 1} C_{2}) ..... (x +^{2 n + 1} C_{n})$ is

$2^{n + 1}$

$2^{2 n + 1} - 1$

$2^{2 n} - 1$

$2^{2 n}$

Q. The coefficient of xn in the polynomial (x+2n+1C0​)(x+2n+1C1​)(x+2n+1C2​).....(x+2n+1Cn​) is

2n+1

22n+1−1

22n−1

22n

Q. The coefficient of $x^{n}$ in the polynomial $(x +^{2 n + 1} C_{0}) (x +^{2 n + 1} C_{1}) (x +^{2 n + 1} C_{2}) ..... (x +^{2 n + 1} C_{n})$ is

$2^{n + 1}$

$2^{2 n + 1} - 1$

$2^{2 n} - 1$

$2^{2 n}$