Question

# In the word ‘ALLAHABAD’ how many of these vowels occupy the even positions?

Hint:Find out the number of vowels and their places and apply Factorial.
First count the vowels in the given word. Later place the vowels as said in the question and find the permutation.

Total no. of vowels in the word ‘ALLAHABAD’ = 4 (all a’s);
Even places of vowels = 2nd, 4th, 6th and 8th;
So, these four places can be occupied by 4 vowels in $\dfrac{{4!}}{{4!}}$ = 1 way;
Now, 5 places are left in which 5 letters (two are same and three different);
Same letter is L and two different letters are B,H and D.
These can be arranged in $\dfrac{{5!}}{{2!}}$ ways;
Total no. of words in which vowels occupy the even places = $\dfrac{{5!}}{{2!}}$
As we know,
$n! = n(n - 1)!$
= $\dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}$= 160

Note:For vowels, space occupied would be 1 but in the case of consonant we are having total of 5 spaces of which two sets we are having in which 1 is of same consonant(L) and the other is of different (B,H,D)
Permutation is the act of arranging objects. Combination is the selection of a few items from the total number of items.