Precalculus, Chapter 9, 9.5, Section 9.5, Problem 51

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

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