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Question: How many 9 letter strings are there that contain at least 3 distinct vowels?

I am studying and I was wondering if this answer could be an alternative answer to the question above: $$\binom{5}{3}\binom{21}{6}+\binom{5}{4}\binom{21}{5}+\binom{5}{5}\binom{21}{4}.$$

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Three distinct vowels? E.g. does "eeeeeeeee" count? – Douglas S. Stones Apr 19 '13 at 17:38
No, they are distinct vowels – Manuel Apr 19 '13 at 17:42
Thanks, I edited in into the question statement, and used LaTeX to encode your answer. – Douglas S. Stones Apr 19 '13 at 17:45
Your count ignores the order of the letters in the string. (And assumes that every letter in the string is distinct.) – ShreevatsaR Apr 19 '13 at 17:50
Is the answer right? – Manuel Apr 19 '13 at 18:01

Your answer is how many ways there are to choose nine distinct letters without repetition that include at least three distinct vowels. It appears the question allows repetition and considers order different.

To get a numeric answer, I would define $R(n)$ to be the number of strings of length $n$ with no vowels, $S(n)$ to be the number of strings of length $n$ with one distinct vowel, $T(n)$ with two and $U(n)$ with three or more. Then we have the base case $R(0)=1,S(0)=T(0)=U(0)=0$ and the recurrences $R(n)=21R(n-1),S(n)=22S(n-1)+5R(n-1),T(n)=23T(n-1)+4S(n-1),U(n)=26U(n-1)+3T(n-1)$ A spreadsheet will make quick work of this.

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