Question

The letters of the word PROMISE are to be arranged so that three vowels should not come together. Find the number of ways of arrangements?
A) 4320
B) 4694
C) 4957
D) 4871

Answer 1

Hi there!
Here's the answer:

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Given,

PROMISE - A 7 letter word

Vowels in the given word : O, I, E
No. of Vowels = 3

¶¶¶ STEP-1 :
Find No. of ways in which 7 letters of the given word can be arranged :

No. of ways 7 letters can be arranged = 7! = 5040 ways


¶¶¶ STEP- 2:
Find No. of ways the vowels can be arranged such that 3 vowels come together:

When the vowels OIE are always together, they are treated as one letter

(OIE) _ _ _ _

Then the Remaining letters PRMS are to be arranged.

Now, 5 letters { (4 consonants +1 vowel group of three treated as single letter) can be arranged in 5! = 120 ways.


Now,
( O I E )
The vowels can be arranged among themselves in 3! = 6 ways

•°• Required No. of ways = 5! × 3! = 720 ways.


¶¶¶ STEP-3:
No. of ways of arrangements so that three vowels should not come together
= No. of ways 7 letters can be arranged - No. of ways the vowels can be arranged such that 3 vowels come together.

•°• Required No. of ways of arrangements = 7! - (5! × 3!)
= 5040 - 720
= 4320 ways.


This answer exists in Option A

•°• Option A is the correct choice.


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