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I'm stuck on this problem:

Consider the five letters A, B, C, D, and E. How many words with four letters can you create if each letter can be used at most two times? (One letter can i.e. be used 0, 1, or 2 times)

At first I thought it would be Permutation(10,4), but that is far off. Any ideas?

Thanks!

N. F. Taussig
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user5744148
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    Discern 3 cases: i) no letters are used twice ii) one letter is used twice iii) two letters are used twice. The cases are mutually exclusive and covering. – drhab Jan 07 '16 at 11:56

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See create cases where we can create $4$ letters is CASE1 all letters distinct and CASE 2 where two different letters are repeated twice. A letter appearing $0$ times and selecting $4$ distinct letters is the same. So for case 1 total ways are $5.4.3.2=120$ and ways where two distinct letters are repeated twice=${5\choose 2}.\frac{4!}{2!.2!}=60$ so total ways are $120+60=180$ CASE 3 where 1 letter is used twice so ways are ${5\choose 1}.{4\choose 2}.4.3=360$ so total ways are $120+60+360=540$ .

Archis Welankar
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  • There is a third case that you do not mention: exactly one letter is used twice. – drhab Jan 07 '16 at 12:03
  • @ drhab i had it in mind but i directly mixed it up with calculations of second case – Archis Welankar Jan 07 '16 at 12:24
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    Exactly one letter used twice gives $360$ possibilities. First choose the letter that is used twice: $5$ posibilities. Then choose the position of these $2$ letters: $\binom42=6$ possibilities. Then choose the letter for the open position at left: $4$ possibilities. Then choose the letter for the remaining open position: $3$ possibilities. – drhab Jan 07 '16 at 14:48
  • But it is repetiyion so $5C1.4!/2!$ – Archis Welankar Jan 07 '16 at 14:49
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    What is repeated then? Or: which possibility is counted more than once? – drhab Jan 07 '16 at 14:50