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I have the answers to the following two questions, but I'm stumped as to why the answers are calculated this way:

Q.1) There are six comics: A, B, C, D, E, F; How many ways are there to select six comics? Answer: C(6+6-1, 6-1)

Q.2) There are 20 balls. 6 red, 6 green, 8 purple. In how many ways can we select five balls if balls of the same color are considered identical? Answer: C(3+5-1, 5)

What I don't understand is why we subtract 1 from 6 in the first question while leaving 5 just the way it is in the 2nd question.

Any help would be appreciated. Thanks!

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up vote 0 down vote accepted

Notice that $C(6+6-1,6-1)=C(6+6-1,6)$, so it doesn't matter whether you subtract 1 from the 6 or not.

In the second problem, $C(3+5-1,5)=C(3+5-1,3-1)$, so you are in fact subtracting 1 --- it's just that you're subtracting it from the 3 (the number of different kinds of item), not from the 5 (the number of items chosen). This is the same as the first problem, where you either subtract 1 from the number of different kinds of item, 6, or don't subtract it from the number of items chosen, which is also 6.

But the best thing to do is to understand where these formulas are coming from.

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Thank you. I understand! – pauliwago Jul 11 '12 at 4:56

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