Olympiad topic · P1–P6
Combinations
Counting arrangements and choices appears in the Hua Xia Cup from the P1 and P2 first rounds to the P5 finals.
Try one
★★ MediumThere are 3 roads from Ming's home to the library and 2 roads from the library to school.
There is also 1 road from home to school that doesn't pass the library.
Ming never doubles back.
How many different ways can Ming walk from home to school?
- Hint 1: Split the ways into two kinds: through the library, and not.
- Hint 2: Through the library: every road on the first leg can be followed by every road on the second.
- 1.Through the library: each of the 3 roads on the first leg can be followed by any of the 2 roads on the second, 3 × 2 = 6.
- 2.Not through the library: 1.
- 3.In all: 6 + 1 = 7.
- ⚠️Common mistake: adding all three, 6.
Answer: 7 ways
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsOne multiplication or one addition.
- ★★MediumFree2 question types · unlimited questionsHandshakes, routes in stages.
- ★★★HardFree2 question types · unlimited questionsNumbers from digit cards, latecomers joining.
- ★★★★Expert (competition level)VIP5 question types · unlimited questionsArrangements with people together or apart, paying with a limited number of coins, numbers with given digits exactly once, passing a ball, sharing out prizes.
Combinations: what it is and how to solve it
Counting the ways to choose, arrange or travel.
For example: 3 ice-cream flavours and 4 cones: one of each, how many choices? 3 × 4 = 12.
The key idea: Steps one after another multiply; separate cases add. For pairs where order doesn't matter (handshakes), each person shakes with everyone else, then halve it, since each handshake was counted twice.
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