To solve this problem, we first analyze the structure of the word "PATALIPUTRA."

Breakdown of the Word:

• Vowels: A, A, I, U, A (5 vowels)
• Consonants: P, T, L, P, T, R (6 consonants)

The Task:

We need to find how many different words can be formed using the letters of the word "PATALIPUTRA" while maintaining the relative order of vowels and consonants.

This means:

• The relative order among vowels must remain the same: A, A, I, U, A.
• The relative order among consonants must remain the same: P, T, L, P, T, R.
• However, the vowels and consonants can be interwoven in any way as long as their respective orders are preserved.

Step 1: Total number of positions

The word "PATALIPUTRA" has a total of 11 letters (5 vowels + 6 consonants).

Step 2: Choosing positions for vowels

Out of these 11 positions, we need to choose 5 positions for the vowels. The remaining 6 positions will automatically be for the consonants.

The number of ways to choose 5 positions from 11 is given by the combination formula:

$\binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462$

Step 3: Conclusion

Thus, the total number of words that can be formed without changing the relative order of vowels and consonants is:

$\boxed{462}$

Let me know if you'd like more details or have any questions! Here are 5 related questions to further explore this concept:

1. How many words can be formed if we consider all letters in "PATALIPUTRA" as unique?
2. What is the probability of selecting a specific arrangement where vowels occupy odd positions?
3. How many arrangements can be made if the first and last letters must be consonants?
4. What happens to the number of arrangements if two of the consonants are swapped in position?
5. Can the vowels ever be adjacent in any of these arrangements?

Tip: When calculating combinations, remember that $\binom{n}{r} = \binom{n}{n-r}$, which can simplify computations.