Question

How many words with or without meaning can be made by using all the letters of the word ENVIRONMENT in which all the vowels are never together.

Answer 1

Dear Student,

Vowels in the word = E , I , O , E (4)
Total number of letters in the word = 11

Now, consider all the vowels as one letter
Therefore total number of letters = 8
Total number of word that can be formed when vowels are together = $\frac{8!}{3!}$ x ⁹C₄ (After arranging letters except vowels there will be 9 gaps where vowels can be inserted
= $\frac{8!}{3!}\times \frac{9!}{4!5!}$
= $\frac{8\times 7\times 6\times 5}{3\times 2\times 1}\times \frac{9\times 8\times 7\times 6}{1}$
= 846720
Regards