In how many ways can the letters of the word INTERMEDIATE be arranged so that: the vowels always occupy even places? the relative order of vowels and consonants do not alter?

Answer

(i) 21600 (ii) 21600

Explanation

(i) The given word 'INTERMEDIATE' has 12 letters, namely 6 vowels and 6 consonants.
Since vowels occupy even places, we have
`N square T square R square M square D square T square`
type of arrangements , where vowels occupy even places, as shown by boxes.
It has 6 consonants namely 2 T's, 1 N, 1 R, 1 M and 1 D.
Number of arrangements 6 consonants `=(6!)/(2!)=360.`
The given word has 6 vowels, 2 I's, 3 E's and 1 A.
Number of arrangements of 6 vowels `=(6!)/((2!)xx(3!))=60.`
`therefore " total number of words formed "=(360xx60)=21600.`
(ii) It is given that we may replace a vowel by another vowel and we can replace a consonant by a consonant, given herewith.
Number of arrangements of consonants `=(6!)/(2!)=360.`
Number of arrangements of vowels `=(6!)/((3!)xx(2!))=60.`
Total number of words formed `=(360xx60)=21600.`