Numerical answer
In how many ways can 15 be written as a sum of three distinct positive integers, if the order of the terms does not matter? (For example, 1 + 2 + 12 and 12 + 2 + 1 count as the same way.)
Show answer and solution
Answer: 12
Find triples x < y < z with x + y + z = 15. Fix the smallest number x:
- x = 1: y + z = 14, 2 ≤ y < z ⇒ y = 2..6 → 5 ways
- x = 2: y + z = 13, y ≥ 3, y < 6.5 ⇒ y = 3..6 → 4 ways
- x = 3: y + z = 12, y = 4, 5 → 2 ways
- x = 4: y + z = 11, y = 5 → 1 way
- x = 5: y + z = 10 with y ≥ 6 needs z ≤ 4 — impossible.
Total = 5 + 4 + 2 + 1 = 12.
Tip: Fixing the smallest part and bounding the middle part (y < (15 − x)/2) is the fastest systematic count.
- x = 1: y + z = 14, 2 ≤ y < z ⇒ y = 2..6 → 5 ways
- x = 2: y + z = 13, y ≥ 3, y < 6.5 ⇒ y = 3..6 → 4 ways
- x = 3: y + z = 12, y = 4, 5 → 2 ways
- x = 4: y + z = 11, y = 5 → 1 way
- x = 5: y + z = 10 with y ≥ 6 needs z ≤ 4 — impossible.
Total = 5 + 4 + 2 + 1 = 12.
Tip: Fixing the smallest part and bounding the middle part (y < (15 − x)/2) is the fastest systematic count.
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