Final answer:
The question involves finding a list of positive integers that sum to 9 for different numbers of summands. One summand is just 9; two summands can be (1, 8), (2, 7), (3, 6), (4, 5); and three summands include combinations like (1, 1, 7), (1, 2, 6), and so on. The meaning of p_{3(9)} is unclear without additional context.
Step-by-step explanation:
The question asks us to find all possible combinations of positive integers that sum to 9 using 1 summand, 2 summands, and 3 summands.
- One summand: 9
- Two summands: (1, 8), (2, 7), (3, 6), (4, 5)
- Three summands: (1, 1, 7), (1, 2, 6), (1, 3, 5), (1, 4, 4), (2, 2, 5), (2, 3, 4), (3, 3, 3)
As for the notation p_{3(9)}, this seems to refer to a partition function p(n,k) where n is the number we want to partition and k is the number of parts in the partition. However, without further context, it's unclear how p_{3(9)} should be interpreted. If this relates to a specific mathematical rule or function not adequately described in the question, it would be necessary to refer to academic resources for the accurate definition.