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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.

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.