Question

How many arrangements are possible with the letter of the word 'profile' if no two consonants are to come together?

Answer 1

In this problem, the letters in the word PROFILE are to be arranged so that no two consonants come together. In the expression in brackets, notice how there are three Xs and 4 underscores [ _X_X_X_]. The Xs represent places where vowels are allowed to go in our allowed words. There 3 places where we are allowed to put vowels. The number of ways these three places can be filled with vowels is 3!=6 ways. The next step in the problem is to decide where the consonants are to go. There are 4 consonants and they have 4 places to go. There are 4 ways to fill the first place, 3 possible ways to fill the second space, three ways to fill the third space, and 1 way to fill the last spot. There is a total of 4!=24 ways to fill the underscores.

From this information, we can gather that there are $(3!)(4!) =(3\times 2\times 1)(4\times\times 3\times 2\times 1)=144$ number of words that meet this condition.