[tex]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.[/tex]
