How many words can be formed with the letters of the word 'PATALIPUTRA' without changing the relative order of the vowels and consonants?

The word 'PATALIPUTRA' has 10 letters in total, with 5 vowels (A, A, I, U, A) and 5 consonants (P, T, L, P, T, R).

Step 1: Arrange the vowels in their relative order. This can be done in only 1 way as we cannot change the relative order of the vowels. So, A, A, I, U, A will always remain in this order.

Step 2: Arrange the consonants in their relative order. This can also be done in only 1 way as we cannot change the relative order of the consonants. So, P, T, L, P, T, R will always remain in this order.

Step 3: We have to insert 5 consonants in the arrangement of 5 vowels. This can be done in 6C5 ways. The 6 possible positions are A_A_I_U_A. We have to choose 5 positions out of these 6 to place the consonants.

So, the total number of words that can be formed with the letters of the word 'PATALIPUTRA' without changing the relative order of the vowels and consonants is 1 (from step 1) * 1 (from step 2) * 6C5 (from step 3) = 6.