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I am not able to think straight and formulate a solution to this problem clearly in my head.

Assume that a vowel is one of the five letters A,E,I,O or U.

(a) How many eleven-letter sequences from the alphabet contain exactly three vowels?

Solution.

Here is my attempt to solve this problem.

One possible eleven letter sequence is, $vcccvccvcc$.

(1) There are ${11\choose3}$ ways to select three positions where the vowels would be placed.

(2) For each placement above, there are $P(5,3)$ ordered lists of vowels of length $3$.

(3) There are $P(21,8)$ ordered lists of consonants of length $8$ for each consonant placement.

The required number of eleven-letter sequences are ${11\choose3}P(5,3){8\choose8}P(21,8)$.

I have made an assumption all along, that repetitions were not allowed. Clearly, this assumption is not true in the context of this question.

a. Is my above solution correct?

b. What if repetitions are allowed?

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If repetition is allowed:

  • Choose $3$ places for vowels: $\binom{11}{3}$
  • Fill those places with vowels: $5^3$
  • Fill the remaining places with non-vowels: $(26-5)^{11-3}$

Hence the number of such sequences is $\binom{11}{3}\cdot5^3\cdot(26-5)^{11-3}$.

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