KA-0035Handshakes and pairs
3 points, difficulty 1 of 3Level 5–6about 45sEight people meet and everyone shakes hands with everyone else exactly once. How many handshakes take place?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Each handshake involves two people.
2The strategy
Count how many hands each person shakes, multiply by the number of people, and then correct for the fact that every handshake has been counted twice, once from each side.
3The full solution
Each of the 8 people shakes 7 hands, giving 8 x 7 = 56, but each handshake was counted twice, so the answer is 56 divided by 2 = 28.
Solution
The reliable way
Each of the 8 people shakes hands with the other 7, which gives 8 x 7 = 56 - but that counts every handshake twice, once from each participant. Halving gives 28. The transferable idea: when each object is counted once from each end, divide by two.
The elegant way
A handshake is a choice of 2 people from 8, and there are 8 x 7 divided by 2 = 28 such pairs.
Why this is on the test: The handshake shape reappears as lines between points and games in a league, and forgetting to halve is the universal error.