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 $\displaystyle ^7C_3=35$ but the answer is 13.
Thanks for any help.
Hello, arze!
Your answer would be correct if there were 7 different letters.How many combinations of three letters taken from the letters A,A,B,B,C,C,D are there?
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: .$\displaystyle {4\choose3} \,=\,4$ 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: .$\displaystyle 3\cdot3 \,=\,9$ combinations.
Therefore, there are: .$\displaystyle 4 + 9 \:=\:13$ possible combinations.