How many words can be formed with the letters of the word ‘PRACTICE’ so that vowels always occupy the even places?

Sure, here is the step by step solution:

1. The word 'PRACTICE' has 8 letters, out of which 3 are vowels (A, I, E) and 5 are consonants (P, R, C, T, C).

2. In an 8-letter word, there are 4 even places (2nd, 4th, 6th, 8th).

3. The 3 vowels can be arranged in the 4 even places in 4P3 ways. The permutation formula nPr = n! / (n-r)! is used here, where n is the total number of options, r is the number of options chosen at a time, and '!' denotes factorial. So, 4P3 = 4! / (4-3)! = 24.

4. The remaining 5 consonants and 1 remaining even place can be arranged in the remaining 5 places in 5P5 ways = 5! / (5-5)! = 120.

5. Therefore, the total number of words that can be formed with the letters of the word ‘PRACTICE’ so that vowels always occupy the even places is 24 * 120 = 2880.