Answer to Question #140389 in Discrete Mathematics for Promise Omiponle

Question

What is the coefficient of x^3 y^13 in the expansion of (−1x+2y)^16?

Answer 1

By Binomial Theorem

$$(a+b)^n=\dbinom{n}{0}a^nb^0+\dbinom{n}{1}a^{n-1}b^1+...+$$

$$+\dbinom{n}{k}a^kb^{n-k}+...+\dbinom{n}{n}a^0b^n$$

We have $a=-x, b=2y, n=16.$

We find the coefficient of $x^3 y^{13}$

$$k=3, \dbinom{n}{k}a^kb^{n-k}=\dbinom{16}{3}(-x)^3(2y)^{16-3}=$$

$$=-\dfrac{16!}{3!(16-3)!}(8192)x^3y^{13}=-560(8192)x^3y^{13}=$$

$$=-4587520x^3y^{13}$$

The coefficient of $x^3y^{3}$ in the expansion of $(-1x+2y)^{16}$ is $-4587520.$