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# The number of different words that can be formed from the letters of the word INTERMEDIATE.Such that two vowels never come together is ________

$\begin{array}{1 1}(A)\;151200\\(B)\;150200\\(C)\;153200\\(D)\;154200\end{array}$

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A)
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• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
Total no of consonants =6
Total no of vowels =6
Total no of letters in the word INTERMEDIATE =12
No of ways of arranging the consonants of which two are alike =$\large\frac{6!}{2!}$
$\Rightarrow \large\frac{6\times 5\times 4\times 3\times 2}{2}$
$\Rightarrow 360$
No of ways of arranging the vowels =$7P_6\times \large\frac{1}{3!}\times \frac{1}{2!}$
$\Rightarrow \large\frac{7!}{1!}\times \frac{1}{3!}\times \frac{1}{2!}$
$\Rightarrow \large\frac{7\times 6\times 5\times 4\times 3!}{3!\times 2}$
$\Rightarrow 420$
Total no of ways =$420\times 360$
$\Rightarrow 151200$
Hence (A) is the correct answer.

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A)

Find the number of permutations of the letters of the word “INTERMEDIATE” such that the vowels always occupy odd places.