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In how many different ways can the letters of the word ‘THERAPIST’ be arranged so that the vowels never come together?

In how many different ways can the letters of the word ‘THERAPIST’ be arranged so that the vowels never come together? Correct Answer 166320

In the word THERAPIST there are 9 letters and T appears twice.

∴ No. of ways to arrange the THERAPIST = 9!/2! = 181440

First we find number of ways when vowels come together.

In the word THERAPIST the vowels are EAI, we treat them as one letter as they are adjacent to each other.

∴ we have ⇒ THRPST(EAI)

In this 7 (6 + 1) letters T appears twice.

∴ Number of ways to arrange them = 7!/2! = 2520 ways

Now, the 3 vowels can also be arranged in different ways.

∴ No. of ways to arrange the vowels = 3! = 6 ways

∴ No. of ways when vowels come together = 2520 × 6 = 15120 ways

∴ Number of ways when vowels never come together,

⇒ 181440 – 15120 = 166320

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