In how many ways can 7 be written as the sum of two different whole numbers bigger than zero? Swapping the order does not make a new way.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Start from the smaller number and count up.
2The strategy
Let the smaller number run 1, 2, 3 and so on. The larger one is forced each time. Stop as soon as the smaller number would overtake the larger, because after that the pairs repeat.
3The full solution
1 + 6, 2 + 5, 3 + 4. The next would be 4 + 3, which repeats the third pair. So there are 3 ways.
Solution— no shortcut on this one
Let the smaller number lead. 1 + 6, 2 + 5, 3 + 4 — and then 4 + 3 is the same pair as 3 + 4, so stop. Three ways. The transferable idea: fixing an order on the pair, such as smaller first, is what stops a list from double counting.
Why this is on the test: Deciding a listing order that cannot repeat itself is the underlying skill; without it, every unordered count comes out doubled.