First slide

Multinomial expansion

Question

The coefficient of x¹⁰in the expansion of ${(1 + x^{2} - x^{3})}^{8}$ is

Moderate

A

476
B

496
C

506
D

528

Solution

We rewrite the given expression as ${[1 + x^{2} (1 - x)]}^{8}$ and expand by using the binomial theorem. We have, ${[1 + x^{2} (1 - x)]}^{8}$
$=^{8} C_{0} +^{8} C_{1} x^{2} (1 - x) +^{8} C_{2} x^{4} (1 - x)^{2} +^{8} C_{3} x^{6} (1 - x)^{3} +^{8} C_{4} x^{8} (1 - x)^{4} +^{8} C_{5} x^{10} (1 - x)^{5} + \dots$
The two terms which contain $x^{10} {are}^{8} C_{4} x^{8} (1 - x)^{8}$ and $^{8} C_{5} x^{10} (1 - x)^{5}$ .
Thus, the coefficient of x¹⁰ in the given expression is given by $^{8} C_{4}$ [coefficient of x² in the expansion of $(1 - x)^{4}] +^{8} C_{3}$
$\begin{array}{l} =^{8} C_{4} (6) +^{8} C_{5} = \frac{8!}{4! 4!} (6) + \frac{8!}{3! 5!} \\ = (70) (6) + 56 = 476 \end{array}$

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