Subjects combinatorics

Handshake Count

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Handshake Count


1. **Problem statement:** If 12 people each shake hands with every other person exactly once, how many handshakes occur in total? 2. **Formula used:** The number of handshakes among $n$ people where each person shakes hands with every other person once is given by the combination formula: $$\text{Number of handshakes} = \binom{n}{2} = \frac{n(n-1)}{2}$$ This formula counts the number of unique pairs that can be formed from $n$ people. 3. **Applying the formula:** Here, $n=12$. $$\text{Number of handshakes} = \frac{12 \times (12-1)}{2} = \frac{12 \times 11}{2}$$ 4. **Simplify the expression:** $$\frac{12 \times 11}{2} = 6 \times 11 = 66$$ 5. **Conclusion:** There will be **66** handshakes in total when 12 people each shake hands with every other person once.