How many 5-letter words contain two vowels and three consonants?

Question

In this question, words are sequences of letters from the English alphabet, vowels are the letters a, e, i, o, and u, and all the other letters are consonants.

1. (a) How many 5-letter words contain two vowels and three consonants?
2. (b) How many 5-letter words with two vowels and three consonants consist of five dif- ferent letters, appearing in alphabetical order?

Answer 1

Hi Luke,

First, let's establish this fact:

There are 5 vowels and 21 consonants in English alphabet letters.

a) How many 5-letter words contain two vowels and three consonants?

Step 1: First lets pick 2 random spots to place those 2 vowels in the 5 letter word:

This can be done in (5 C 2) ways.

Step 2: Once the spots for the vowels have been determined, lets count the number of ways we can start filling in with letters:

vowel spot 1 - has 5 choices

vowel spot 2 - has 5 choices

consonant spot 1 - has 21 choices

consonant spot 2 - has 21 choices

consonant spot 3 - has 21 choices

so the 5 letters can be filled in the 5 spots in the following number of ways:

5 * 5 * 21 * 21 * 21 = (5^2 * 21^3)

Step 3: Multiply the result from step 1 and step 2 to get the total number of 5-letter words containing two vowels and three consonants

So the answer is (5 C 2) * (5^2 * 21^3) = 2315250

b) How many 5-letter words with two vowels and three consonants consist of five different letters, appearing in alphabetical order?

Key observation:

We need five different letters, appearing in alphabetical order. If you take a specific combination of the 5 different letters, for example, "defabc". Although there are 5! different arrangements, we only need 1 where all letters are arranged in alphabetical order, which is "abcdef".

So, effectively, we only need to count how many different combinations of 5 different letter words containing 2 vowels and 3 consonants exist!

This can done as follows:

Select 2 different vowels from 5 - can be done in (5 C 2) ways

Select 3 different consonants from 21 - can be done in (21 C 3) ways

So the answer is (5 C 2)*(21 C 3) = 13300

Hope this helps!