Answer to Question #293425 in Discrete Mathematics for Tege

Question

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 ?

Answer 1

There are 21 vowels.

We have $21!$  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$^{22}P_5={22!\over17!}$

By multiplication rule, the total number of orderings in which no two vowels occur consecutively equals, ${22!\times 21!\over 17!}$.