# The number of words that can be made by re-arranging the letters of the word APURBA

Question:

The number of words that can be made by re-arranging the letters of the word APURBA so that vowels and consonants are alternate is

(a) 18

(b) 35

(c) 36

(d) none of these.

Solution:

(c) 36

The word APURBA is a 6 letter word consisting of 3 vowels that can be arranged in 3 alternate places, in $\frac{3 !}{2 !}$ ways.

The remaining 3 consonants can be arranged in the remaining 3 places in 3! ways.

$\therefore$ Total number of words that can be formed $=\frac{3 !}{2 !} \times 3 !=18$

But this whole arrangement can be set-up in total two ways, i.e either  VCVCVC or CVCVCV.

$\therefore$ Total number of words $=18 \times 2=36$