Answer to Question #142108 in Discrete Mathematics for Promise Omiponle

Question

Find the coefficient of x^7 y^9 in the expansion of (4x+ 5y)^16.

Answer 1

Expansion of $(a+b)^n$ gives us $(n+1)$ terms which are given by binomial expansion $\dbinom{n}{r}a^{(n-r)}b^r$ , where $r$ ranges from $n$ to 0.

Note that powers of $a$ and $b$ add up to $n$ and in the given problem this $n=7+9=16$.

In $(4x+5y)^{16}$ , we need coefficient of $x^7y^9$ , we have $7^{th}$ power of $x$ and as such $r=16-7=9$

and as such the desired coefficient of $x^7y^9$ is given by

$\dbinom{16}{9}(4x)^{(16-9)}(5y)^9=\dfrac{16!}{9!(16-9)!}(4x)^7(5y)^9=11440*16384x^7*1953125y^9= 3.6608E14x^7y^9$

So6 the coefficient is 3.6608E14