How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions ?

Concept:

The number of combinations of n different thing taken k at a time is:

\(^nC_k=\frac{n!}{k!(n-k)!}\) where 0 ≤ k ≤ n

Calculation:

The word 'TIGER' contains 2 vowels I and E and 3 consonants T, G, R.

If the vowels should not occupy the even position means even places should be occupied by the consonants only. 

To fill the even places by the consonants, select any 2 consonants out of 3 and arrange them.

Selection of any 2 consonants out of 3 = ³C₂

The arrangement of these two consonants is given by 2!

Number of ways to fill even places = 3C2 × 2! = 6 

Now, fill the 3 odd places with the rest of the consonants and vowels.

Number of ways to fill odd places = 3! = 6

The total number of ways = 3C2 × 2! × 3! = 6 × 6 = 36

∴ The required number of permutations = 36.