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We a homework question that asks us to find the 3 letter arrangement of the word "Silly". Here is the exact question.

How many three-letter arrangements are there of the letters taken from the word SILLY?

What I did was do $5P3/2!$ and I ended up getting 30. The answer is 33. What am I doing wrong?


I have just realized, should I be adding back 3 because of a permutation that does have the LL's in it?

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When you choose three letters out of 5, one of the possibilities is S,I,L which has no repetitions. So dividing by $2!$ is wrong. – Isomorphism Dec 14 '12 at 10:36
What should be the correct step then? – Jeel Shah Dec 14 '12 at 10:37
up vote 2 down vote accepted

In this small problem the easy way is to note first that there are $4\cdot3\cdot2=24$ ways to pick three different letters and permute them. Then observe that there are $3$ ways to pair LL with one of the other letters, and $3$ ways to place that odd letter relative to the two L’s, so there are $3\cdot3=9$ more $3$-letter arrangements that include both L’s, for a total of $24+9=33$ arrangements.

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