Question

Q.9. How many different words can be formed of the letters of the word

‘FRACTION’ so that :(i) Vowels remain together.​

Answer 1

Answer:

• ACTION
• ION
• ACT
• RAN
• FAN
• RAT
• FAT
• CAT

Answer 2

Answer:

“f r a c t i o n”

No. of vowels = 3.

No. of consonants = 5.

Now, here we can think consonants as the separators/walls between the vowels because the condition says “no two vowels be together”.

So, any word made with the letters of word “fraction”, will look like this…

(V)C(V)C(V)C(V)C(V)C(V)

where, C is a consonants, C belongs to { f , r , c , t ,n }.

and among the above 6 V’s 3 are vowels [ a , i , o] and rest are empty.

Now the problem becomes….

How many ways the 3 places can be chosen out of 6 places to put these 3 vowels in that positions. Answer is easy, 6C3.

How many ways 5 consonants can be arranged with in themselves. Again answer is easy, 5!.

How many ways 3 vowels can be arranged with in themselves. Answer is 3!.

So, Actual answer will be multiplied value of 6C3, 5! and 3! (as above three actions are independent of each other).

6C3 X 5! X 3! = 20 X 120 X 6 = 14,400.

keep sharing knowledge….

Thank you :)