Question:
How many words can be formed out of the letters of the word ‘ORIENTAL’ so that the vowels always occupy the odd places?
Solution:
To find: number of words formed
Condition: vowels occupy odd places
There are 8 letters in the word ORIENTAL and vowels are 4 which are O, I, E,A respectively.
There is 4 odd places in which 4 vowels are to be arranged.
The rest 4 letters can be arranged in $4 !$ Ways.
Formula:
Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is
$P(n, r)=n ! /(n-r) !$
Therefore, the total arrangement is
$P(4,4) \times 4 !=\frac{4 !}{(4-4) !} \times 4 !=\frac{4 !}{0 !} \times 4 !=\frac{24}{1} \times 24=576$
Therefore, total number of words formed are 576