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

What I did below is wrong. Can someone tell me the reason and the correct solution?

$$\left(\binom{3}{2}\times\binom{6}{2}\right) + \left(\binom{3}{3}\times\binom{6}{1}\right)=22$$

AvZ
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Wong
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    For homework questions, you will probably get more responses if you demonstrate that you tried to solve it on your own first. – Peter Webb Mar 01 '15 at 11:51

1 Answers1

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When you take 3 Gs then there are 4 possible options with the 4 other letters in the word:T,E,R and A

and when you take only 2 Gs there are 6 letters left that you can choose from:

A1, A2, E1, E2, R and T.

you can't simply take 6C2 because it will count for example G,G,A1,T as different from G,G,A2,T which is not true. so you have to subtract the instances when one of the As or one of the Es are with another letter in the 4 letter word as they are counted twice.

(A1 with R, E1, E2, T, and E1 with R, A2, T) a total of 7 instances.

6C2 - 7 = 15-7 = 8

8 (2Gs)+4 (3Gs)=12

It may be easier to simply list the possible words, but this is one way you could think about it.

vink007
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