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  4. The co-efficient of x7 in the expansion of (1 - x4) (1 + x)9 is

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Q. The co-efficient of $x^7$ in the expansion of $(1 - x^4)$ $(1 + x)^9$ is

Binomial Theorem

Solution:

$\left(1-x^{4}\right) (1+\,{}^{9}c_{1}x+\,{}^{9}c_{2}x^{2}+\,{}^{9}c_{3}x^{3}+\,{}^{9}c_{4}x^{4}+\,{}^{9}c_{5}x^{5}$
$+\,{}^{9}c_{6}x^{6}+\,{}^{9}c_{7}x^{7}+\,{}^{9}c_{8}x^{8}+\,{}^{9}c_{9}x^{9} )$
Co-eff. of $x^{7} = \,{}^{9}c_{7}-\,{}^{7}c_{3} = \,{}^{7}c_{2} - \,{}^{9}c_{3}$
$= \frac{9\times8}{1\times 2} - \frac{9\times 8\times 7}{1\times 2\times 3}$
$ = 36 \left[1-\frac{7}{3}\right]$