Question

What is the coefficient of the x^6y^3 term in the expansion of (x+2y)^9?

Answer 1

use the binomial expansion formula for (a+b)^n
(sorry, cannot type it here, please google)
n!/[(n-k)!k!]a^(n-k)b^k
x^6y^3 is when k=3
in this case, n=9
so the coefficient when k=3 is: 9!/[(9-3)!3!]*(2)^3=
`(9*8*7)/(3*2*1) *2^3=672`

Answer 2

The coefficient of
`x^6y^3` is 672.

Explanation:

By the binomial expansion,

`(a+b)^n=\sum_(r=0)^n ^nC_r a^(n-r) b^r`

Where,

`^nC_r=(n!)/(r!(n-r)!)`

Thus,

`(x+2y)^9=\sum_(r=0)^n ^9C_r x^(9-r) (2y)^r`

For finding the coefficient of the
`x^6y^3`

r = 3,

Hence, the term in which
`x^6y^3` is present is,

`^9C_3 x^6 (2y)^3`

`=84* x^6* 8y^3`

`=672x^6y^3`

Therefore, the coefficient of
`x^6y^3` is 672.