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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.
6
9
16
24
27
256
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:
We need to find:
This is a permutation problem. When we have n distinct objects to arrange in n positions, there are \(\mathrm{n!}\) ways to do it.
We have 4 remaining tones to place in 4 positions (9th through 12th):
Total ways = \(\mathrm{4 \times 3 \times 2 \times 1 = 24}\)
Once the 9th tone is chosen, we have 3 remaining tones to place in 3 positions (10th through 12th):
Total ways = \(\mathrm{3 \times 2 \times 1 = 6}\)
Final Answer:
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.