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UPSEE 2018 Solved Paper

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Question : 131 of 150

Marks: +1, -0

The coefficient of

x^{5}

in the expansion of

(1+x^{2})^{5}(1+x)^{4}

is

40
50
60
$-50$

Solution:

Coefficient of

x^{5}

in the product

(1+x^{2})^{5}(1+x)^{4}

=

The coefficient of

x^{5}

in

[{}^5C_{0}+{}^5C_{1}(x^{2})+{}^5C_{2}(x^{2})^{2}+{}^5C_{3}(x^{2})^{3}

+{}^5C_{4}(x^{2})^{4}+{}^5C_{5}(x^{2})^{5}]

\times[{}^4C_{0}+{}^4C_{1}x+{}^4C_{2}x^{2}+{}^4C_{3}x^{3}

+{}^4C_{4}x^{4}]

=

The coefficient of

x^{5}

in

[{}^5C_{0}+{}^5C_{1}x^{2}+{}^5C_{2}x^{4}+{}^5C_{3}x^{6}+{}^5C_{4}x^{8}

+{}^5C_{5}x^{10}]

\times[{}^4C_{0}+{}^4C_{1}x+{}^4C_{2}x^{2}+{}^4C_{3}x^{3}+{}^4C_{4}x^{4}]

={}^5C_{1}\times{}^4C_{3}+{}^5C_{2}\times{}^4C_{1}

=5\times4+10\times4

=20+40=60

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