Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.
Solution
To solve this problem, we need to consider two cases: selecting 4 letters with exactly 2 Gs and selecting 4 letters with exactly 3 Gs.
Case 1: Exactly 2 Gs
We choose 2 Gs from the 3 available Gs. The remaining 2 letters must be chosen from the letters A, E, R, T. The possible combinations are:
- GG with AA
- GG with AE
- GG with EE
- GG with RA
- GG with RE
- GG with RT
- GG with TA
- GG with TE
This gives us 8 ways.
Case 2: Exactly 3 Gs
We choose 3 Gs from the 3 available Gs. The remaining letter must be chosen from A, E, R, T. The possible combinations are:
- GGG with A
- GGG with E
- GGG with R
- GGG with T
This gives us 4 ways.
Total
\(Adding both cases together, we have a total of 8 + 4 = 12 ways.\)
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