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Not totally sure if I'm understanding the questions correctly:

Consider the permutations of the set of 26 letters of the English alphabet

  • How many total permutations are possible? P(26,26)=26!
  • How many permutations begin with a? P(26,25)
  • How many permutations begin with z and end with a? P(26,24)
  • How many permutations begin with the 5 vowels (in any order), which are followed by the remaining consonants (in any order)? I have no idea.
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  • $\begingroup$ So in the last one, there are 21 consonants to be permuted. How many ways can that be done ? ... And there are 5 vowels you can choose for the beginning, this can be done in 5! ways. Multiply the two out and you have your answer as 5! times 21!. Thanks @DavidQuinn. I have corrected the error $\endgroup$ – Shailesh Sep 13 '15 at 17:45
  • $\begingroup$ Do you mean 5! @Shailesh? $\endgroup$ – David Quinn Sep 13 '15 at 17:46
  • $\begingroup$ Ohh okay. That makes sense. Thank you! And are my other answers right? $\endgroup$ – ematth7 Sep 13 '15 at 17:53
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There is only one way to put a in the first spot. Then, how many ways are there to arrange the remaining 25 letters in 25 spots? $25!$

There is only one way to put z first and 1 way to put a last. Then, how many ways are there to arrange the remaining 24 letters in 24 spots? $24!$

How many ways can we arrange the 5 vowels among five spots? $5!$ Then how many ways can we arrange the remaining 21 letters in 21 spots? $21!$ So the total ways to do this are $5!*21!$

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