A hard question on combinations for P4–P5, with a step-by-step solution, diagrams and the answer. Counting arrangements and choices appears in the Hua Xia Cup from the P1 and P2 first rounds to the P5 finals.
The question
At a birthday party, every two guests shook hands once:
36 handshakes in all.
Then 2 more guests arrived.
Each newcomer shook hands once with everyone there, other newcomers included.
How many more handshakes were there after the newcomers arrived?
Show the hints (3)
💡Hint 1: First find how many guests there were: how many people shaking hands in pairs make exactly 36?
💡Hint 2: Newcomers shake hands with two groups: the guests already there, and each other.
💡Hint 3: There were 9 guests. Each newcomer shakes hands with each of them; the newcomers also shake hands in pairs.
Worked solution
Have a go first, then read the solution step by step.
1.9 × 8 ÷ 2 = 36, so there were 9 guests.
2.Newcomers with the guests: 9 each, 2 × 9 = 18.
3.Newcomers among themselves: 2 × 1 ÷ 2 = 1.
4.Extra handshakes: 18 + 1 = 19.
5.Check: 11 people make 55, and 55 − 36 = 19. Common mistake: forgetting the newcomers' handshakes with each other.
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