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Q)

The number of words which can be formed out of the letters of the word ARTICLE,so that vowels occupy the even place is

$\begin{array}{1 1}(A)\;1440\\(B)\;144\\(C)\;7!\\(D)\;4C_4\times 3C_3\end{array} $

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A)
Toolbox:
  • $C(n,r)=\large\frac{n!}{r!(n-r)!}$
  • $n!=n(n-1)(n-2)(n-3)......(3)(2)(1)$
In word ARTICLE.There are 3 vowels and 4 consonants
Total no of letters =7
Total no of even place =3
There are 3 vowels to be filled in 3 places
Hence the no of ways =$3C_3=1$
The vowels can arrange among themselves in 3! ways
$\Rightarrow 6$
Now the four consonants can fill the remaining 4 places in =4!
$\Rightarrow 4\times 3\times 2\times 1$
$\Rightarrow 24$
The total no of words formed =$1\times 6\times 24$
$\Rightarrow 144$ ways
Hence (B) is the correct answer.
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