eight students from different countries meet to plan an international peace ceremony. each student shakes the hand of each other student. how many handshakes are there altogether?

There are 28 handshakes altogether.

How to see it

Each of the 8 students shakes hands with every other student once.
If you let n=8n=8n=8, the total number of handshakes when everyone meets everyone once is given by:

Handshakes=n(n−1)2\text{Handshakes}=\frac{n(n-1)}{2}Handshakes=2n(n−1)​

Here:

8×72=562=28\frac{8\times 7}{2}=\frac{56}{2}=2828×7​=256​=28

You can also think of it as:

• Student 1 shakes hands with 7 others
• Student 2 shakes with 6 new people (not counting Student 1 again)
• Student 3 shakes with 5 new people, and so on

So you add:

7+6+5+4+3+2+1=287+6+5+4+3+2+1=287+6+5+4+3+2+1=28

Answer: 28 handshakes.