ConverterHub

Handshake Calculator

How many handshakes for n people?

Result

66

How it is calculated

The tool counts the handshakes when each person in a group shakes hands with every other person exactly once: H = n × (n − 1) / 2, where n is the number of people. Each of the n people shakes n − 1 hands, which counts every handshake twice (once from each side), so the total is halved. This is the same as choosing 2 people from n, also written C(n, 2). n must be a whole number of at least 2: 0, 1, negatives and fractions give no result. The same formula counts games in a round-robin league or links between n computers. The result is an integer with thousands separators.

Worked example

At a meeting of 12 people everyone shakes hands once with everyone else. Each person shakes 11 hands, so 12 × 11 = 132 hand-clasps are counted, but each handshake is counted from both sides. Halve it: 132 / 2 = 66. The calculator shows 66 handshakes. With 2 people there is exactly 1 handshake, which matches 2 × 1 / 2.

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Common mistakes

  • Forgetting to halve and reporting n × (n − 1), which counts every handshake twice.
  • Using n × n / 2, as if people shook their own hand. Each person has only n − 1 possible partners.
  • Counting yourself separately. If you are one of the n people, include yourself in n.
  • Entering the number of handshakes per person instead of the number of people.
  • Applying it to a case where pairs may repeat or where only some people meet. The formula assumes every pair meets exactly once.

About this calculator

The classic handshake problem asks how many total handshakes occur if everyone in a room of n people shakes hands with everyone else exactly once. The answer is n(n−1) ÷ 2, the same formula as nCr with r = 2, choosing 2 people out of n to form each unique pair, since every handshake corresponds to exactly one unordered pair of people.

This same choose-2 structure underlies counting the number of games in a round-robin tournament where every team plays every other team once, the number of edges in a complete graph with n vertices in graph theory, and the number of possible one-to-one connections in a network, which is why it is a staple introductory combinatorics problem used to build intuition before moving to general nCr calculations.

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