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How many different words can be formed from the letters of the

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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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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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Text solutionVerified

No. of letters

= 12; 6

vowels

(2 l, 3 E, 1 A)

and 6 consonants

(2 T, 1 R, 1 M, 1 N, 1 D)

.
Total words

= \frac{12 !}{2 ! 3 ! 2 !}

Gap Method:
No. of ways of arranging 6 consonants (2 alike) is

\frac{6 !}{2} = 360

....(1)

There will be 7 gaps in which 6 vowels if all different can be arranged in

^{7} P_{6}

ways but a 2 are alike of one kind and 3 of other kind.

∴

Number of arranging the vowels is

^{7} P_{6} \cdot \frac{1}{3 ! 2 !} = \frac{7 !}{12} = \frac{5040}{12} = 420

....(2)

Hence the total number of ways when the two vowels never come together by fundamental theorem is

360 \times 420 = 151200

by (1), (2)

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Question Text	How many different words can be formed from the letters of the word INTERMEDIATE? In how many of them, two vowels never come together?
Answer Type	Text solution:1
Upvotes	150