First slide

Multinomial expansion

Question

The coefficient of $x^{2} y^{3}$ in the expansion of $(1 - x + y)^{20}$ is

Moderate

A

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

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

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

none of these

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 $x^{2} y^{3}$ , we have the term $\frac{20!}{15! 2! 3!} 1^{15} (- x)^{2} (y)^{3}$
Hence, the coefficient of $x^{2} y^{3} is \frac{20!}{15! 2! 3!}$

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