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When creating homework for my students, I came up with this: How many lists of length $5$ can be made from the set $$ \{A,B,C,D,E,F,G,H,I\} $$ if we cannot repeat a letter and they must be in alphabetical order?

Now, the way I would solve this would be to do a case by case analysis depending on the first letter. So, start with $A$ and count the lists by choosing a second letter and so on. Then do the same for $B$ the first letter. My question is whether there is a much less computationally long answer or an answer that is more instructive.

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up vote 3 down vote accepted

There are $9$ letters. Choose $5$.

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Wow. My brain does not always go to the easiest solution. – Joe Johnson 126 Dec 5 '12 at 23:12

The number of such lists is simply $\binom95$: there’s an obvious bijection between them and $5$-letter subsets of the base set.

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