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, ${22!\times 21!\over 17!}$.