what is the numbers of ways to order the 26 letters of the alphabet so that no two of the vowels a,e,i,o,u occur consecutively ?
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There are 21 vowels.
We have ways of ordering these consonants.
There are a total of 22 valid locations for placing 5 vowels. Thus the number of ways of placing the 5 vowels in 5 of the 22 locations is
By multiplication rule, the total number of orderings in which no two vowels occur consecutively equals, .