Question

a)
How many different arrangements can be made from the letters of the word
SIGNIFICANTLY in such a way that:
(1) all the vewels are together.
(17) all the vowels are not together.
OR​

Answer 1

all the vowels are together.  = 12 * 10! =  4,35,45,600‬

all the vowels are not together =  131 * 10! =  47,53,72,800‬

Step-by-step explanation:

SIGNIFICANTLY

Vowels = I , I , I , A    - 4    ( I repeated thrice)

Consonants -  S , G , N  F , C , N , T , L Y  -  9    ( N repeated twice)

Lets find total Possible Words

13!/(3! * 2!)

= 51,89,18,400‬

all the vowels are together.

=> Taking all Vowels as 1  Letter

these 4 vowels can be arranged in   4!/3!  = 4 Ways

Now Total 10 Letters

can be arranged in 10!/2!    Ways

Total ways = 4! * 10!/2!

= 12 * 10!

= 4,35,45,600‬

all the vowels are together.  = 12 * 10! =  4,35,45,600‬

all the vowels are not together = 51,89,18,400‬ -  4,35,45,600‬

= 47,53,72,800‬

13!/(3! * 2!) -  12 * 10!

= 13 * 12 * 11 * 10! / 12  - 12 * 10!

= 143 * 10!  - 12 * 10!

= 131 * 10!

all the vowels are not together =  131 * 10! =  47,53,72,800‬

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