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

^{16} C_{9}

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^{17} C_{8}

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^{16} C_{8}

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Solution

The correct option is A $^{16} C_{9}$
$x^{6}$ in $[(1 + x)^{6} + (1 + x)^{7} + \dots + (1 + x)^{15}$

This is in form of G.P.

Where there are 10 terms

So $(1 + x)^{6} [\frac{(1 + x)^{10} - 1}{x}]$

$\frac{(1 + x)^{16} - (1 + x)^{6}}{x}$

Now coefficient of $x^{7}$ in $(1 + x)^{16} - (1 + x)^{6}$

$^{16} C_{7} \equiv^{16} C_{9}$

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The coefficient of x6 in [(1+x)6+(1+x)7+....+(1+x)15] is

The correct option is A 16C9x6 in [(1+x)6+(1+x)7+…+(1+x)15This is in form of G.P.Where there are 10 termsSo (1+x)6[(1+x)10−1x](1+x)16−(1+x)6xNow coefficient of x7 in (1+x)16−(1+x)616C7≡16C9

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