Understanding the Problem
The word "PARNECIOUS" consists of 10 letters, with vowels and consonants. The vowels in the word are A, E, I, O, U, and must remain in the order A, E, I, O, U.
Identifying Letters
- Total letters: 10 (P, A, R, N, E, C, I, O, U, S)
- Vowels: A, E, I, O, U (5 vowels)
- Consonants: P, R, N, C, S (5 consonants)
Arranging Consonants and Vowels
1. Total Arrangements Without Restrictions
The total arrangements of the letters without any restrictions are calculated as the factorial of the total letters:
- Total arrangements = 10!
2. Fixing the Vowel Order
Since the vowels must remain in the order A, E, I, O, U, we treat the 5 vowels as fixed in their positions. We can place the 5 consonants in the remaining slots.
Calculating Arrangements
- The arrangement of consonants (5 consonants) can be done in:
- 5! ways
- We need to choose 5 positions for the consonants from the 10 available positions. This can be done using combinations:
- Choose 5 positions from 10 = 10 choose 5 = 10! / (5! * 5!)
Final Calculation
- Therefore, the total arrangements respecting the vowel order can be calculated as:
- Total arrangements = (10 choose 5) * (5!) = (10! / (5! * 5!)) * (5!)
- Simplifying this gives:
- Total arrangements = 10! / 5!
Conclusion
Thus, the total number of ways to arrange the letters of the word "PARNECIOUS" without changing the order of the vowels is 2520.