Question

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

Answer 1

`a = 180`.

Explanation:

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

The formula for the nth term in a binomial expansion `(a+ b)^n` is given by

`t_(k + 1) = color(white)(two)_nC_ka^(n - k)b^k`

We have

`k + 1 = 3`

`k = 2`

Use the formula now.

`t_3 = color(white)(two)_10C_2x^(10 -2)(-2y)^2`

The value of `color(white)(two)_10C_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 = 45x^8 4y^2`

`t_3 = 180x^8y^2`

Therefore, `a= 180`.

Hopefully this helps!