In how many arrangements of the word ‘GOLDEN’ will the vowels neve

1.	In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?
Answer» 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) = \(\frac{5!}{(5-5)!}\times\frac{2!}{(2-2)!}\) = \(\frac{5!}{0!}\times\frac{2!}{0!}\) = 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.

In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?

Answer»

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) = \(\frac{5!}{(5-5)!}\times\frac{2!}{(2-2)!}\)

= \(\frac{5!}{0!}\times\frac{2!}{0!}\) = 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.