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

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

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.