`(10x - 3y)^12 , n = 10` Find the nth term in the expansion of the binomial.

In order to solve this problem we can use the following equation of the binomial theorem: 

`(a+b)^n =sum_(k=1)^n ((n!)/((n-k)! *k!)) * a^(n-k) * b^k`

where: 

a= first term

b= last term

n = exponent (power in original equation) 

k = term required - 1 

This will be clearer by solving the above example: 

Our example is as follows:

`(10x - 3y)^12`  

In this example we are looking for the 10th term, therefore: 

`a = 10x`

`b=3y`

`n =12`

`k = 10 - 1 = 9` (the 10th term we are looking for so we subtract one from it)

Because we are looking for the 10th term the following equation will be used: 

`t_(k+1) =(n!)/((n-k)! *k!) * a^(n-k) * b^k`

`t_(9+1) = (12!)/((12-9)!(9)!) (10x)^(12-9) * (3y)^9`

`t_10 = 220 (1000x^3)(19683y^9)`

`t_10 = 4330260000 x^3 y^9`

So if we expand our binomial, and if we are looking for the 10th term, our answer is `4330260000x^3y^9`