Question

Find the number of ways of permuting the letters of the word $PICTURE$ so that
(i) All vowels come together
(ii) No two vowels come together
(iii) The relative positions of vowels and consonants are not disturbed

Answer 1

Rearrange the word PICTURE as PCTRIUE

i)

To keep the vowels together consider IUE as a single unit and the no. of permutation in IUE is 3!

total units become P,C,T,R,IUE = 5

so permutations will be 5!

so total permutations will be 5!*3!

ii)

No two vowels should be together so GIVE GAPS around them like I_U_E

Permutate IUE in 3! ways.

we need two fill in that we will choose from remaining in 4C2 ways

and will consider the selected two and IUE as a single unit and arrange with the remaining 2, so arranging three units is 3!ways

so total ways are - 5C2*3!*3! = 5!*3 = 360ways.

iii)

The relative position is not altered so we need to arrange 3 vowels in 3! ways and 4 consonants in 4! ways give total to be

4!*3!