Question

In how many ways can 4 consonants and 2 vowels be selected in the English alphabet consisting of 21 consonants and 5 vowels?

Answer 1

See a solution process below:

Explanation:

If consonants and vowels can be picked just one time each, the number of ways 4 consonants and 2 vowels can be selected is:

For the first consonant, there are 21 possible consonants to select from.

For the second consonant, there are 20 possible consonants to select from.

For the third consonant, there are 19 possible consonants to select from.

For the fourth consonant, there are 18 possible consonants to select from.

For the first vowel, there are 5 possible vowels to select from.

For the second vowel, there are 4 possible vowels to select from.

This means there are;

`21 xx 20 xx 19 xx 18 xx 5 xx 4 = 2,872,800` ways to select these letters

Answer 2

Unordered: there are 59,850 unique six-tuples that have 4 consonants and 2 vowels.

Ordered: there are 43,092,000 "words" with 4 consonants and 2 vowels

(All letters are unique.)

Explanation:

If order doesn't matter (e.g. we're just choosing unique letters to form something like a Scrabble rack), then there are `""_21C_4` ways to select 4 consonants, and `""_5C_2` ways to select 2 vowels.

Consonants: `((21),(4))=(21xx20xx19xx18)/(4xx3xx2xx1)=5985`

Vowels: `((5),(2))=(5xx4)/(2xx1) = 10`

Since each group of 4 consonants could be paired with each group of 2 vowels, we multiply these two numbers together:

4 Consonants & 2 Vowels: `((21),(4))xx((5),(2))="59,850".`

If order does matter, (e.g. we're looking to see how many "words" we can form with these 4 consonants and 2 vowels), then each of the 59,850 groups can be permuted in `6!` ways:

Number of "words":

`59850 xx 6! = "59,850" xx 720 = "43,092,000"`