Question

Find the number of the words that can be formed from the letters of the word ARTICLE if vowels always occupy the even places.

Answer 1

3   vowels   A E I  
4  consonants   R T C L 

vowels occupy even places - 2, 4, 6 
consonants occupy  places 1, 3, 5, 7

3 vowels occupy 3 places    in   3 !   = ³P₃

4 consonants occupy 4 places.  in  ⁴P₄ = 4! ways

There are no repetitions of any letter. all letters are different from others.

The arrangements of vowels is independent from arrangement of consonants. So the total number is the multiplication of both.

So the number of words =  3 ! * 4 ! = 24 * 6 = 144

Answer 2

see the attachment for analysis soln.