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Coefficient of {x^6} in {( {1 + x} )^6} + {( {1 + x} )^7} + ..

Question

Coefficient of $x^{6}$ in $(1 + x)^{6} + (1 + x)^{7} + .... + (1 + x)^{15}$ is

^{16} C_{10}

^{16} C_{8}

^{16} C_{9}

^{16} C_{12}

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Coefficient of

x^{6}

(1 + x)^{6} + (1 + x)^{7} + .... + (1 + x)^{15}

15 mins ago

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15 mins ago

Text solutionVerified

Coeff of

x^{6}

a) (1 + x)^{6}

^{6} C_{6} = 1

b) (1 + x)^{7}

^{7} C_{6}

(1 + x)^{15}

^{15} C_{6}

\Rightarrow

Coefficient of

x^{6}

(1 + x)^{6} + (1 + x)^{7} + - - - - - (1 + x)^{15}

= 1 +^{7} C_{6} +^{8} C_{6} + - - - - - - +^{15} C_{6}

=^{7} C_{7} +^{7} C_{6} +^{8} C_{6} + - - - - - +^{15} C_{6}

=^{8} C_{7} +^{8} C_{6} +^{9} C_{6} - - - - - +^{15} C_{6}

=^{9} C_{7} +^{9} C_{6} + - - - - [∵^{n} C_{r} + n_{r - 1} =^{n + 1} C_{r}]

=^{16} C_{7}

=^{16} C_{9}

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