# Math Help - Combinations

1. ## Combinations

I hope this is the right place to put this thread.
How many combinations of three letters taken from the letters A,A,B,B,C,C,D are there?
I take it as $^7C_3=35$ but the answer is 13.
Thanks for any help.

2. Hello, arze!

How many combinations of three letters taken from the letters A,A,B,B,C,C,D are there?
Your answer would be correct if there were 7 different letters.
Because of the duplicated letters, I counted the cases rather primitively.

How many 3-letter combinations have 3 different letters?

There are 4 different letters available, and we pick 3 of them.
. . There are: . ${4\choose3} \,=\,4$ combinations.

How many 3-letter combinations have 2 of one letter and 1 other letter?

There 3 choices for the pair (A, B, or C).
Then there are 3 choices for the single letter.
. . There are: . $3\cdot3 \,=\,9$ combinations.

Therefore, there are: . $4 + 9 \:=\:13$ possible combinations.