Question

In how many different ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together?

• A. 720
• B. 1440
• C. 5040
• D. 3600
• E. 4800

Answer 1

The correct option is D 3600

Total number of letters is 7 and these letters can be written in 7! ways
$=7\times 6\times 5\times 4\times 3\times 2\times 1$
= 5040 ways
There are seven letters in the word THERAPY including 2 vowels (E,A) and five consonants (THRPY).
Now, consider two vowels as one letter.
We have 6 letters which can be arranged in ${}^6{P}_6$ ways = 6! ways
But vowels can be arranged in 2! ways. Hence, the number of ways that all vowels will come together = 6 ! $\times$ 2!
$=1\times 2\times 3\times 4\times 5\times 6\times 2$
= 1440
Total number of ways in which vowels will never come together = 5040 - 1440
= 3600