Question

In how many differents ways can the letter of the word 'professional'be arranged in such a way that the vowels always come together?

Answer 1

Total number of vowels- o,e,i,o,a = 5 ( 1 repeat)

professional contains 12 letters it can be grouped as p,r,f,s,s,n,l, (o,e,I,o,a) as 8 units. vowels are considered as 1 unit here

To arrange these 8 units total possible ways= 8!/2! (as s repeats 2 times)

And the vowels can be arranged among themselves in 5!/2! ways ( o repeats Twice)


so total no of ways is 8!*5!/4 = 12,09,600

Answer 2

Good question.

The word 'professional' contains 1 p, 1 r, 2 o, 1 f, 1 e, 2 s, 1 i, 1 n, 1 a, and 1 l.

Here the vowels are 2 o, 1 e, 1 i, and 1 a. i. e., a total of 5 vowels.

As the vowels always come together, the arrangement should be like,

vvvvvccccccc

cvvvvvcccccc

ccvvvvvccccc

cccvvvvvcccc

ccccvvvvvccc

cccccvvvvvcc

ccccccvvvvvc

cccccccvvvvv

(v indicates vowels and c indicates consonants.)

8 types of arrangements are there.

The no. of arrangements of 5 vowels with 2 same will be 5! ÷ 2! = 60.

The no. of arrangements of 7 consonants with 2 same will be 7! ÷ 2! = 2520.

∴ The answer is 60 x 2520 x 8 = 1209600.

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