First slide

Multinomial expansion

Question

The coefficient of x²y³ in the expansion of $(1 - x + y)^{20}$ is

Easy

A

$\frac{20!}{2! 3!}$
B

$- \frac{20!}{2! 3!}$
C

$\frac{20!}{5! 2! 3!}$
D

$\frac{20!}{15! 2! 3!}$

Solution

The general term in the expansion of $(1 - x + y)^{20}$ is
$\frac{20!}{r! s! t!} 1^{r} (- x)^{s} (y)^{t}, where r + s + t = 20$
For x²y³, $r = 15, s = 1, t = 3$ we have the term
$\frac{20!}{15! 2! 3!} 1^{15} (- x)^{2} (y)^{3}$
Hence, the coefficient of x²y³ is
$\frac{20!}{15! 2! 3!}$

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