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.