Step One
Develop Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1 <<<< How the 5th power expands.
Step Two
Confirm that nCr gives the same thing as the triangle gives.
The third term is outlined as 10 a^3 b^2 The powers must add to 5.
The 10 can be found this way
nCr Where n is the row number and r is the term you want so
n = 5
r = 3
nCr = n! / [(n - r)! r!]
5C3 = 5! / ((3!) (2!) )
5C3 = 5 * 4 / 2
5C3 = 10 Just as the triangle predicted.
Step Three
Find a^3 b^2
a = 3x
b = - 5y
a^3 = (3x)^3
a^3 = 27x^3
b^2 = (-5y)^2
b^2 = 25 y^2
Step Three
Find the Answer
Third term = 10*27x^3*25y^2
Third term = 6750 x^3y^2
