Question

Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.

Answer 1

AGGREGATE

there are 9 elements total, and when we break then into common sets:

2: AA
3: GGG
1: R
2: EE
1: T

we have a general counting rule that we can apply to this: given n elements, and multiple parts (a,b,c,...,k):
\[\frac{n!}{a!~b!~c!~...~k!}\]

Answer 2

so im thinking, if we remove the Gs to start with we have:\[\frac{(9-3)!}{2!\cancel{3!}1!2!1!}\]distinct ways to organize the other parts.

Answer 3

but thats as far as my thoughts have taken me so far