Illuminations: Handshakes

# Handshakes

When the nine justices of the Supreme Court meet, they greet one another with a handshake to show harmony of purpose, even though their views may differ. How many handshakes occur? In general, how many handshakes occur when there are n people?

### Instructions

 The Show People buttons will show dots for 2‑12 people on the screen. After one of these buttons is pressed, a Show Handshakes button will appear; when the Handshakes button is pressed, a line segment will be drawn to represent each handshake that occurs, and the total number of handshakes will be placed in the table. The Reset Drawing button clears the picture. The Reset Chart button clears the numbers from the table.

### Exploration

 Click Show 2 People. Then click Show Handshakes with 2 People. How many handshakes are there? Click Show 3 People. Then click Show Handshakes with 3 People. The number of people was increased by 1; by how many was the number of handshakes increased? Now, drag the dot next to the numeral 3 in the drawing. As the point moves, the lines representing the new handshakes will also move. Repeat for 4, 5, … 12 handshakes. Each time, note the increase in the number of handshakes. What rule can you use to describe the increase in the number of handshakes? What rule will allow you to predict the number of handshakes when there are n people?