Question

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

Answer 1

Hint: To solve this question we have to understand that there are 7 letters in the word ARTICLE and notice that we have second, fourth and sixth that means three even places and also A , I , E three vowels so we rearrange them through every possible way, also arrange remaining letter on remaining place through 4!.

Complete step-by-step answer:

We have ARTICLE
Here vowels letters are A , I and E and consonant letters are R , T , C , L.
ARTICLE is a 7 letter word so we have to fill seven places in which three are even which are second , fourth and sixth places and in these three places we have to fill vowel letters A ,I and E and here four odd places which are first , third, fifth , and seventh place and we have to fill consonants R , T , C , L .
So we have three places for three vowels and four places for four consonants
Hence the number of ways of arranging three vowels at three places is 3!
And the number of ways of arranging four consonants at four places is 4!.
By fundamental law of multiplication the total number of words formed from the letters of word ARTICLE where vowels occupy even places is $3! \times 4! = 144$
Here we use the first 3! For arrangement of vowels we have to also arrange consonants for making words and come so we use sign of multiplication if or comes then we use sign of addition and 4! For arrangements of consonants.

Note: Whenever we get this type of question the key concept of solving is we have to have knowledge of permutation and combination that means where we have to use sign of multiplication and where addition and also care about rearrangements of words so that all the words can be counted.