Question

The coefficient of x^6 in {(1+x)^6+(1+x)^7+...(1+x)^15} is ??

Answer 1

`BINOMIAL \: \: \: THEORAM \\ \\ \\ Given \: expression \: is \: G.P. \: with \: \\ first \: \: term \: \: \: {(1 + x)}^(6) \: and \: \: \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: common \: ratio \: (1 + x). \\ \\\\ Expression \: - \\ \\ \frac{ {(1 + x)}^(6)(1 - {(1 + x)}^(10)) }{1 - (1 + x)} \\ \\ i.e. \: sum \: of \: 10 \: terms \: of \: G.P. \\ \\ = \frac{{(1 + x)}^(16) - {(1 + x)}^(6) }{x} \\ \\ \\ Therefore \: , \: the \: required \: coefficient \: - \\ \\ \\ = \: the \: coefficient \: of \: \: {x}^(7) \: \: in \: \\ \: \: \: \: \: \: \: \: \: \: ( {(1 + x)}^(16) - {(1 + x)}^(6) ) \\ \\ \\ \\ = \: \: \: {}^(16) C_(7) \: \: = \: \: {}^(16) C_(9) \: \: \: \: \: \: \: \: \: \: \: Ans.`