How many words can be formed out of the letters of the word banana so that the consonants occupy the even places?

Solution : In word ARTICLE, there are `3` vowels and `4` consonants.
Total number of letters = `7`
Total number of even place =`3`
There are `3` vowels to be filled in `3` places.
Hence, the number of ways = `^3C​_3`​=`1`
The vowels can arrange among themselves in `3!=6 `ways.
Now, the `4` consonants can fill the remaining 4 places in `4!=24` ways.
Therefore, total number of ways = `1×6×24=144` ways.

The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is 144.

Explanation:

Total number of letters in the ‘ARTICLE’ is 7 out which A, E, I are vowels and R, T, C, L are consonants

Given that vowels occupy even place

∴ Possible arrangement can be shown as below

C, V, C, V, C, V, C i.e. on 2nd, 4th and 6th places

Therefore, number of arrangement = 3P3 = 3! = 6 ways

Now consonants can be placed at 1, 3, 5 and 7th place

∴ Number of arrangement = 4P4 = 4! = 24

So, the total number of arrangements = 6 × 24 = 144.

How many words can be formed out of the letters of the word 'ARTICLE', so that vowels occupy even places?

Solution

The word ARTICLE consists of 3 vowels, which have to be arranged in 3 even places. This can be done in 3! ways.
Now, the remaining 4 consonants can be arranged in the remaining 4 places in 4! ways.
∴ Total number of words in which the vowels occupy only even places = 3!\[\times\]4! = 144

Concept: Factorial N [N!] Permutations and Combinations

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Solution

The correct option is B

144

The word ARTICLE consists of 3 vowels that have to be arranged in the three even places. This can be done in 3! ways.

And, the remaining 4 consonants can be arranged among themselves in 4! ways.

∴ Total number of ways = 3!×4!=144

How many words can be formed out of the letters of the word ARTICLE so that vowels occupy the even places?

Answer

Verified

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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