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5 vowels in the alphabet
There are five vowels. Notice that any of the vowels could be chosen from 0 though 5 times in any given combination. We therefore need the number of combinations of 5 from 5 _with replacement_. This is the number ( 5 + 5 - 1 ) choose 5 = 126.
5 Vowels in the English Alphabet.
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There are 3 consonants from 21 and 2 vowels from 5. That gives 21C3 * 5C2 combinations = 21*20*19/(3*2*1) *5*4/(2*1) = 1330*10 = 13300 combinations in all.