In how many ways can the letters of the word “INTERMEDIATE” be arranged so that :

i. the vowels always occupy even places?

ii. the relative order of vowels and consonants do not alter?

Given the word INTERMEDIATE. IT has 12 words out of which 6 are vowels, and 6 of them are consonants.

(i) the vowels always occupy even places?

i.e., There are 6 vowels, and there can occupy even places, i.e. position number 2, 4, 6, 8, 10, and 12.

A number of ways of arranging 6 vowels among 6 places = 6!/(2! x 3!)

(Since there are 2 repeating vowels I (twice) and E (thrice)).

A number of ways of arranging 6 consonants among 6 places = 6! / 2!

(Since there are 1 repeating consonant T repeating twice).

Total number of ways of arranging vowels and consonants such that vowels can occupy only even positions

= 21600

Hence, a number of ways of arranging INTERMEDIATE’s letters such that vowels can occupy only even positions is equals to 21600.

(ii) The relative order of vowels and consonants does not alter?

A number of ways of arranging vowels = 6! / (2! x 3!)

A number of ways of arranging consonants = 6! / 2!

Total number of ways of arranging the letter of word INTERMEDIATE such that the relative order of vowels and consonants does not alter

= 21600

Hence, a total number of ways of arranging the letters of the word INTERMEDIATE such that the relative orders of vowels and consonants do not change is 21600.