In a party 15 people shake their hands with each other. How many times did the hand-shakes take place?

### Concept & Formula When every person in a group shakes hands with every other person exactly once, the total number of handshakes is a basic combination problem. The number of unique pairs that can be formed from $n$ items is given by: $$ \text{Total Handshakes} = \frac{n(n - 1)}{2} $$ ### Step-by-Step Solution * **Given:** The total number of people, $n = 15$. * **Calculation:** Substitute $n = 15$ into the combination formula. * Total Handshakes = $\frac{15 \times (15 - 1)}{2}$ * Total Handshakes = $\frac{15 \times 14}{2}$ * Total Handshakes = $15 \times 7 = 105$ ### Exam Strategy & Shortcut For handshake problems, do not waste time writing the formula. Immediately multiply the given number by the number just below it, and halve it. Mentally calculating $15 \times 7 = 105$ takes less than 3 seconds. ### Common Pitfall A very common mistake is using $n(n - 1)$ which gives 210. This happens when students confuse handshakes (where A shaking B's hand is the same as B shaking A's hand) with gift exchanges (which are directional). Always divide by 2 for handshakes. ### Final Answer **Therefore, the correct answer is 105.**