A composer is writing a melody using the twelve-tone technique of musical composition. In the twelve-tone technique, each of the...
GMAT Two Part Analysis : (TPA) Questions
A composer is writing a melody using the twelve-tone technique of musical composition. In the twelve-tone technique, each of the twelve tones of the musical scale must be used before any of the tones are repeated. In the composer's melody, no two tones will be used simultaneously. The composer has already chosen the first 8 tones of his melody following this technique.
Based on the information provided, and assuming the composer uses the technique described, select for Ways to order 4 tones the number of ways the composer can order the remaining 4 tones of his melody, and select for Ways to order 3 tones the number of ways, once the composer has chosen the ninth tone of his melody, the composer can order the remaining 3 tones of his melody. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visualization
Let's create a simple visual to represent the twelve-tone melody:
Positions: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Status: ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ? ? ? ?
(Already chosen - 8 tones) (Remaining 4 tones)
Key Information:
- Total tones available: 12 (each must be used exactly once)
- Tones already chosen: 8
- Remaining tones to place: 4
- No two tones used simultaneously (linear ordering)
Phase 2: Understanding the Question
We need to find:
- Ways to order 4 tones: The number of ways to arrange the 4 remaining tones in positions 9-12
- Ways to order 3 tones: After choosing which tone goes in position 9, the number of ways to arrange the remaining 3 tones in positions 10-12
Mathematical Principle
This is a permutation problem. When we have n distinct objects to arrange in n positions, there are \(\mathrm{n!}\) ways to do it.
Phase 3: Finding the Answer
Part 1: Ways to order 4 tones
We have 4 remaining tones to place in 4 positions (9th through 12th):
- For position 9: 4 choices
- For position 10: 3 choices (one tone already used)
- For position 11: 2 choices (two tones already used)
- For position 12: 1 choice (only one tone left)
Total ways = \(\mathrm{4 \times 3 \times 2 \times 1 = 24}\)
Part 2: Ways to order 3 tones
Once the 9th tone is chosen, we have 3 remaining tones to place in 3 positions (10th through 12th):
- For position 10: 3 choices
- For position 11: 2 choices (one tone already used)
- For position 12: 1 choice (only one tone left)
Total ways = \(\mathrm{3 \times 2 \times 1 = 6}\)
Phase 4: Solution
Final Answer:
- Ways to order 4 tones: 24
- Ways to order 3 tones: 6
These values make mathematical sense: arranging 4 distinct objects gives us \(\mathrm{4! = 24}\) permutations, while arranging 3 distinct objects gives us \(\mathrm{3! = 6}\) permutations.