The coefficient of xn in the polynomial (x+nC0)(x+3. nC1)(x+5. nC2) .... (x+(2n+1)nCn) is

Question

The coefficient of xn in the polynomial (x+nC0)(x+3. nC1)(x+5. nC2) .... (x+(2n+1)nCn) is (A) n . 2n (B) n . 2n + 1 (C) (n + 1) . 2n (D) (n + 1) . 2n+1

Tardigrade Admin · Accepted Answer

(x+nC0)(x+3. nC1)(x+5. nC2) .... (x+(2n+1).nCn) = xn+1+xn nC0+3. nC1+5. nC2+..... +(2n+1).nCn +..... Coeff. of xn = nC0 +3. nC1+5. nC2+..... +(2n+1).nCn = 1+ (nC1 +2. nC1)+(nC2+4. nC2) + ....+(nCn+2n. nCn) = (1+nC1+..... + nCn)+2(nC1 + 2nC2 + .... +n. nCn) = 2n +2[n+2. (n(n-1)/2!)+3. (n(n-1)(n-2)/3!)+...+ n.1] = 2n +2n [1+n-1C1+n-1C2+ ..... +n-1Cn-1] = 2n+2n. 2n-1 = 2n (1+n) = (n+1) . 2n

Q. The coefficient of $x^{n}$ in the polynomial
$(x +^{n} C_{0}) (x + 3 .^{n} C_{1}) (x + 5 .^{n} C_{2}) .... (x + (2 n + 1)^{n} C_{n})$ is

$n . 2^{n}$

$n . 2^{n + 1}$

$(n + 1) . 2^{n}$

$(n + 1) . 2^{n + 1}$

Q. The coefficient of xn in the polynomial (x+nC0​)(x+3.nC1​)(x+5.nC2​)....(x+(2n+1)nCn​) is

n.2n

n.2n+1

(n+1).2n

(n+1).2n+1

Q. The coefficient of $x^{n}$ in the polynomial
$(x +^{n} C_{0}) (x + 3 .^{n} C_{1}) (x + 5 .^{n} C_{2}) .... (x + (2 n + 1)^{n} C_{n})$ is

$n . 2^{n}$

$n . 2^{n + 1}$

$(n + 1) . 2^{n}$

$(n + 1) . 2^{n + 1}$