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

i) 302400 different arrangements can be made when all vowels are together.

ii) 181650 different arrangements can be made when all the vowels are not together.

Step-by-step explanation:

a) Vowels in the word SIGNIFICANTLY are three I's, one A.

Total vowels = 3 + 1 = 4

Consonants in the word SIGNIFICANTLY are S, G, two N's, F, C, T, L, Y.

Total consonants = 1 + 1 + 2 + 1 + 1 + 1 + 1 +1 = 9

1) All vowels are together. Take all vowels as 1 then add with consonants.

Total letters = 1 + 9 = 10

Number of arrangements = $\frac{10!}{3! 2!}$ = $\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3!}{3!2!}$ = 302400

[Since 3! for three I's, 2! for two N's]

Hence, 302400 different arrangements can be made when all vowels are together.

2) All the vowels are not together. First arrange the consonants then arrange the vowels.

There are 9 consonants. We can arrange 9 consonants in $\frac{9!}{2!} = \frac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2!}{2!}$ = 181440 ways

After arranging 9 consonants there 10 places will remain blank.

Now arrange 4 vowels in 10 places = $10_{C_{4} } = \frac{10\times 9\times 8\times 7\times 6!}{6!\times 4\times 3\times 2}$ = 210 ways

Total number of arrangements = 181440 + 210 = 181650

Hence, 181650 different arrangements can be made when all the vowels are not together.