To find: number of words

Condition: vowels should never occur together.

There are 6 letters in the word GOLDEN in which there are 2 vowels.

Total number of words in which vowels never come together =

Total number of words – total number of words in which the vowels come together.

A total number of words is 6! = 720 words.

Consider the vowels as a group.

Hence there are 5 groups that can be arranged in P(5,5) ways, and vowels can be arranged in P(2,2,) ways.

Formula:

Number of permutations of n distinct objects among r different places, where repetition is not allowed, is

P(n,r) = n!/(n-r)!

Total arrangements = P(5,5) × P(2,2) = ×

= × = 120 × 2 = 240.

Hence a total number of words having vowels together is 240.

Therefore, the number of words in which vowels don’t come together is 720 – 240 = 480 words.