### 1

There are 10 people in a room. Each person shakes hand with 3 other people, what is the total number of handshakes?

A

Note: we don’t care if person A shakes with B, C, D or E, F, G. But every person reaches out three times. Thus, the total number of hand reach-outs is $$ 10 * 3 = 30 $$ Every 2 reach-outs forms 1 handshake, thus total number of handshakes is 30 / 2 = 15.

To generalize such question, X people to shake hands with others, the maximum number of handshakes is: $$ \frac{X\cdot(X-1)}{2} $$

### 2

Every two companies participating in a trade sign 1 contract. There are 45 signed contracts. How many companies are there at least?

A

Let \(X\) be the number of companies. \(X\) can sign with at most \(X-1\) other companies. Thus \(\frac{X\cdot(X-1)}{2} = 45 \Rightarrow X = 10\). At least 10 companies are needed to sign 45 contracts.

### 3

Given 7 soccer teams. Each team competes with all other 6 teams 3 times. How many games in total?

A

There are total 7 teams. Each team needs to play 3 * 6 = 18 games. Each game has 2 teams involved. Thus: $$ \frac{7 \times 18}{2} = 63 $$