Given: the word is ‘STRANGE.’

To find: a number of arrangements in which vowels never come together

To find out these, we will find the total number of arrangements irrespective of any condition and subtract those arrangements in which vowels come together

Total number of letters in the word = 7

Formula used:

Number of arrangements of n things taken all at a time = P(n, n)

∴ Total number of arrangements irrespective of any condition

= the number of arrangements of 7 things taken all at a time

= P(7, 7)

{∵ 0! = 1}

= 7!

= 7 × 6 × 5 × 4 × 3 × 2 × 1

= 5040

Number of vowels in this word = 2(A, E)

Now, consider these two vowels as one entity(AE together as a single letter)

So, the total number of letters = 6(AE S T R N G)

Formula used:

Number of arrangements of n things taken all at a time = P(n, n)

∴ Total number of arrangements

= the number of arrangements of 6 things taken all at a time

= P(6, 6)

{∵ 0! = 1}

= 6!

= 6 × 5 × 4 × 3 × 2 × 1

= 720

Two vowels which are together as a letter can be arranged in 2

Ways like EA or AE

∴ Total number of arrangements in which vowels come together = 2 × 720 = 1440

Hence, the total number of arrangements in which vowel never come together = 5040 – 1440 = 3600