# How do you find the coefficient of a of the term ax^8y^2 in the expansion of the binomial (x-2y)^10?

Feb 16, 2017

$a = 180$.

#### Explanation:

If we include the 0th term, then there will be 11 terms in this expansion. This means that the term $a {x}^{8} {y}^{2}$ will be the third term (since 11- 8 = 3).

The formula for the nth term in a binomial expansion ${\left(a + b\right)}^{n}$ is given by

${t}_{k + 1} = {\textcolor{w h i t e}{t w o}}_{n} {C}_{k} {a}^{n - k} {b}^{k}$

We have

$k + 1 = 3$

$k = 2$

Use the formula now.

${t}_{3} = {\textcolor{w h i t e}{t w o}}_{10} {C}_{2} {x}^{10 - 2} {\left(- 2 y\right)}^{2}$

The value of ${\textcolor{w h i t e}{t w o}}_{10} {C}_{2}$ can be computed using the formula color(white)(two)_nC_r = (n!)/((n - r)!r!). Therefore, color(white)(two)_10C_2 = (10!)/(8!2!) = 45

${t}_{3} = 45 {x}^{8} 4 {y}^{2}$

${t}_{3} = 180 {x}^{8} {y}^{2}$

Therefore, $a = 180$.

Hopefully this helps!