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Question

How many different words can be formed from the letters of the word INTERMEDIATE? In how many of them, two vowels never come together?

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Solution

No. of letters =12;6 vowels (2l,3E,1A) and 6 consonants (2T,1R,1M,1N,1D).
Total words =12!2!3!2!
Gap Method:
No. of ways of arranging 6 consonants (2 alike) is
6!2=360 ....(1)
There will be 7 gaps in which 6 vowels if all different can be arranged in 7P6 ways but a 2 are alike of one kind and 3 of other kind.
Number of arranging the vowels is
7P613!2!=7!12=504012=420 ....(2)
Hence the total number of ways when the two vowels never come together by fundamental theorem is 360×420=151200 by (1), (2)

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