Question

Find the number of different 8-letter arrangements that can be made from the letters of the word EDUCATION so that all vowels do not occur together.

(a) 40320

(b) 37440

(c) 1440

(d) 2880

I have been asked this question in examination.

The origin of the question is Permutations-2 topic in division Permutations and Combinations of Mathematics – Class 11

Answer 1

Correct choice is (b) 37440

To explain: There are 5 vowels in word EDUCATION. 5 vowels can be arranged in ^5P5 i.e. 5! ways. When all vowels are together, 5 vowels together form one letter and remaining 3 letters i.e. together 4 letters can be arranged in ^4P4 i.e. 4! ways. Total possible arrangements are 5! * 4! = 120*24 = 2880.

So, when all vowels do not occur together, total possible arrangements = 8! – 2880 = 40320 – 2880 = 37440.