What-is-the-coefficient-x-10-in-the-expansion-of-1-x-2-x-3-8-

Question Number 132382 by liberty last updated on 13/Feb/21

What is the coefficient x^{10}

in the expansion of {(1 + x^{2} - x^{3})}^{8}

Answered by EDWIN88 last updated on 13/Feb/21

by multinomial theorem; coefficient of x^{10}

we get from 2 cases

(1) {(1)}^{3} {(x^{2})}^{5} {(- x^{3})}^{0} with coefficient (\begin{matrix} 8 \\ 3, 5, 0 \end{matrix})

= \frac{8!}{3! . 5! . 0!} = \frac{8.7.6}{6} = 56

(2) {(1)}^{4} {(x^{2})}^{2} {(- x^{3})}^{2} with coefficient (\begin{matrix} 8 \\ 4, 2, 2 \end{matrix})

= \frac{8!}{4! . 2! . 2!} = \frac{8.7.6.5}{4} = 420

totally = 476

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