The coefficient of $x^{7}$ in the expansion of $\left(1-x-x^{2}+x^{3}\right)^{6}$ is

JEE Main · 2011 · 10 Apr 2011 · Offline · Binomial Theorem

The coefficient of

x^{7}

in the expansion of

(1 - x - x^{2} + x^{3})^{6}

Answer: (B)

Factor the base as

1 - x - x^{2} + x^{3} = (1 - x) (1 - x^{2})

, so the expression is

(1 - x)^{6} (1 - x^{2})^{6}

. Expanding,

(1 - x)^{6} = \sum_{k} (k 6) (- x)^{k}

and

(1 - x^{2})^{6} = \sum_{j} (j 6) (- x^{2})^{j}

, and the coefficient of

x^{7}

collects the terms with

k + 2 j = 7

, namely

(k, j) = (5, 1), (3, 2), (1, 3)

. These give

(5 6) (- 1)^{5} (1 6) (- 1) + (3 6) (- 1)^{3} (2 6) + (1 6) (- 1) (3 6) (- 1)^{3} = 36 - 300 + 120 = - 144

, so the coefficient of

x^{7}

- 144

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option