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I am very confused about this question.

You have a set of 5 letters {a,b,c,d,e} and you need the number of four letter strings that do not contain aa in the middle.

This is 5 choose 4, but if you are choosing less items than the number that exist and all are distinct how can you possibly end up with a "aa" anywhere?

Dre_Dre
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  • I think every $4$ letter word with letters from the alphabet ${a,b,c,de}$ is fair game, for example $aaaa$ – Asinomás Aug 04 '16 at 03:25
  • Where in the question does it ever say anything about choosing 4 distinct items? It just says "four-letter strings". The letters are distinguished, in the sense that they are not sitting unordered in a bag but one can refer to the first letter or the second letter as specific parts of the string. But why do you think they are distinct? Just because they could be chosen distinctly (because $4 < 5$) doesn't mean that they are! – Erick Wong Aug 04 '16 at 04:48
  • The start of the list: aaba, aabb, aabc, aabd, aaca, aacb, aacc, aacd, aada, aadb, aadc, aadd, abaa, abab, abac, abad, abba, abbb, abbc, abbd, abca, abcb, abcc, abcd, acaa, ... Note that each letter may be used multiple times and there are many more than just $\binom{5}{4}=5$ arrangements. Approach via multiplication principle and/or inclusion exclusion. – JMoravitz Aug 04 '16 at 05:30

2 Answers2

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There are $5^2-1$ options for the two letters in the middle and then $5^2$ options for the letters in the edges. So there are $(5^2-1)\times5^2=24\times 25=600$ words.

Asinomás
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  • I am trying to understand this, can you explain why 5^2 -1 and how you end up with 5^4? – Dre_Dre Aug 04 '16 at 03:29
  • @Dre_Dre There are five choices for each entry. Hence, if there were no restrictions, we could fill the two middle entries in $5^2$ ways. Since we are not allowed to use aa, the number of ways we can fill the two middle entries is $5^2 - 1$. – N. F. Taussig Aug 05 '16 at 09:03
  • What does the 1 represent? – Dre_Dre Aug 05 '16 at 13:34
  • It represents the question for which you give the answer to problem $1$, so for example the sequence $1,2,3,4$ meanas you got it all correct. – Asinomás Aug 05 '16 at 13:49
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Unrestricted, you have $5$ choices for each place in the string, hence $5^4$ strings.

By "middle" I take it to be the central two, so strings like $-aa-$ are banned.
The other $2$ places could be filled in $5^2$ ways,

hence valid number of strings $= 5^4 - 5^2$

true blue anil
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