Question

In how many di淪�erent ways can the letters of the word ‘therapy' be arranged so that the vowel never come together?

Answer 1

RAINBOW consists of 7 distinct letters, in which there are 3 vowels and 4 consonants.

We have to find the total number of arrangements such that the vowels are never together. It could only be possible if consonants and vowels are placed alternatively.

Let us arrange the 4 consonants such that there is a slot of1 l letter between ever consecutive consonants.

_ C _ C _ C _ C _

There are 5 alternate slots appear in the representation but we have only 3 vowels to fit in those 5 slots.

3 slots from 5 empty slots can be selected in 5C3 =105C3 =10 ways

3 vowels can be arranged in those 3 selected slots in 3P3 =3! =63P3 =3! =6 ways

Consonants shown as C in the representation can be arranged among themselves in 4P4 =4! =244P4 =4! =24 ways

Therefore, the total number arrangements of letters of the word RAINBOW such that vowels are never together = 5C3⋅ 3P3⋅ 4P4 =10∗3!∗4! =1440 5C3⋅ 3P3⋅ 4P4 =10∗3!∗4! =1440 ways

I hope it helps!