# Permutations with 26 letters

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.
• 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 – Shailesh Sep 13 '15 at 17:45
• Do you mean 5! @Shailesh? – David Quinn Sep 13 '15 at 17:46
• Ohh okay. That makes sense. Thank you! And are my other answers right? – ematth7 Sep 13 '15 at 17:53

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 $5!*21!$