Tip:A–D to answerE for explanationV for videoS to reveal answer
The coefficient of xn in the expansion of (1 - 2x + 3x^2 - 4x^3 + ... to ∞ - n) is
- A.
n!(n−1)!(2n)!
- B.
[(n−1)!]2(2n)!
Correct Answer: (n!)2(2n)!
Explanation
Concept:
(x+y)n=nC0xn+nC1x(n−1).y+nC2x(n−2).y2+...+nCnyn
(1+x)−2=1−2x+3x2−4x3+...
Calculation:
Here, [1 - 2x + 3x^2 - 4x^3 + ... to ∞]^n
= ((1+x)−2)−n
= (1+x)(2n)
= 2nC0x(2n)+2nC1x(2n−1).(1)1+...2nCnx(2n−n)+...+2nC2n(x)0(1)2n
So, coefficient of xn=2nCn=(2n)!/(n!)2
Hence, option (3) is correct.
Explanation
Concept:
(x+y)n=nC0xn+nC1x(n−1).y+nC2x(n−2).y2+...+nCnyn
(1+x)−2=1−2x+3x2−4x3+...
Calculation:
Here, [1 - 2x + 3x^2 - 4x^3 + ... to ∞]^n
= ((1+x)−2)−n
= (1+x)(2n)
= 2nC0x(2n)+2nC1x(2n−1).(1)1+...2nCnx(2n−n)+...+2nC2n(x)0(1)2n
So, coefficient of xn=2nCn=(2n)!/(n!)2
Hence, option (3) is correct.
