Answer to Question #235714 in Statistics and Probability for Tyza

Question

How many ways are there to make a 3-letter word (not necessarily a proper word) consisting of 2 vowels followed by a consonant, such that all letters are different? HINT: There are 26 letters in the alphabet of which 5 are vowels and the remaining letters are consonants.

Answer 1

The first two letters of the word consisting of two different vowels (hence we select vowels without replacement and order of vowels also matter in the selection therefore we will use permutation for getting answer) and third letter of the word consist of one consonant.

There are 5 vowels and 21 consonants in the alphabet.

$P^n_r = \frac{n!}{(n-r)!}$

Therefore, total number of ways to make 3-letter word $= (P^5_2) \times (P^{21}_1)$

$= \frac{5!}{3!} \times \frac{21!}{20!} \\ = 5 \times 4 \times 21 = 420$

Hence, total number of ways to make 3 letter word is 420.