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I'm asked the following two questions: How many Strings of 6 letters contain

a) Exactly one Vowel

b) At least one Vowel

a) This is what I know: The English alphabet has 21 consonants and 5 vowels. So we choose one of those vowels as such:

$6\choose1$$5\choose1$$21\choose1$$^5$, shouldn't this give me my answer? I'm not exactly sure how to wrap my head around such questions, the answer I'm given is : 122523030

Any help would be much appreciative.

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  • $\begingroup$ Here are 7 words, contradicting your statement: bbbbba, bbbbbe, fffffe, efffff, accccc, daddnn, mmammm $\endgroup$ – Improve Dec 7 '15 at 1:10
  • $\begingroup$ You are correct that $6\choose1$ is in fact the number of ways to chose exactly where in your word the vowel should be placed. Now your next job is to figure out how many different constants you can have at each place and how many different vowels. Perhaps it is easier to try the case where your word has only 2 letters first? $\endgroup$ – Improve Dec 7 '15 at 1:17
  • $\begingroup$ I've updated my answer for a), they're being multiplied $\endgroup$ – TTEd Dec 7 '15 at 1:19
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    $\begingroup$ And that answer is correct! Now a hint for b is that If you know how many 6-letter words there are, and also know how many of them only have consonants, then you also know how many have at least 1 vowel. Why? $\endgroup$ – Improve Dec 7 '15 at 1:21
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    $\begingroup$ Well woudn't it just be the subtraction of the two? $26\choose1$$^6$ - $21\choose1$$^6$ $\endgroup$ – TTEd Dec 7 '15 at 1:23

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