How many words can be formed out of the letters of the word ARTICLE so that vowels occupy the even places?
Answer
Hint: Find the number of vowels and consonants in the word ‘ARTICLE’. Arrange them in even and odd places as there are a total 7 letters in the word. So arrange them in 7 places and find how the 7 letters can be arranged.
Complete step-by-step answer:
Consider the word given ‘ARTICLE’.
Total number of letters in the word ARTICLE = 7.
Number of vowels in the word = A, I and E =3.
Number of consonants in the word = R, T, C and L = 4.
There are 7 places amongst
which the vowel has to be arranged in an even place.
We have to arrange the vowels [A, I, and E] in even places.
It can be done in \[3!\] ways.
We have to arrange the consonants [R, T, C, and L] in odd places. It can be done in \[4!\] ways.
\[\therefore \]We can arrange the 7 letters in \[\left[ 3!\times 4! \right]\]ways.
\[\left[ 3!\times 4! \right]=\left[ 3\times 2\times 1 \right]\times \left[ 4\times 3\times 2\times 1 \right]=6\times 24=144\].
\[\therefore \] We
can form 144 words with the letters of the word ARTICLE where vowels occupy the even places and consonants the odd places.
Note: If the question was asked to arrange the consonants, we will choose the 4 consonants at odd places in\[4!\] ways and then arrange vowels.
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Solution
The correct option is C
576
There are 3 even places in the 7 letter word ARTICLE.
So, we have to arrange 4 consonants in these 3 places in 4P3 ways.
And the remaining 4 letters can be arranged among themselves in 4! ways,
∴ Total number of ways of arrangement =
4P3×4!=4!×4!=576
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