Q. 274.3( 3 Votes )

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?

Answer :

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.


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