At a party of 20 quests, every guest shakes every other guest's hand once. How many handshakes are exchanged during the party? Note that when two people shake hands with each other it counts as one handshake. A 19 B 20 C 39 D (D) 190 E (E) 380

## Step 1<br />This problem is a classic example of a combinatorial problem, specifically a handshake problem. The problem asks us to find the number of handshakes that can be made in a group of 20 people, where each person shakes hands with every other person exactly once.<br /><br />## Step 2<br />The formula to calculate the number of handshakes in a group of \( n \) people is given by the combination formula \( C(n, 2) \), which represents the number of ways to choose 2 people out of \( n \) to shake hands.<br /><br />### **The formula for combinations is \( C(n, k) = \frac{n!}{k!(n-k)!} \)**<br /><br />## Step 3<br />In this case, \( n = 20 \) and \( k = 2 \), so we need to calculate \( C(20, 2) \).<br /><br />## Step 4<br />Substituting the values into the formula, we get:<br /><br />### **\( C(20, 2) = \frac{20!}{2!(20-2)!} = \frac{20!}{2! \cdot 18!} = \frac{20 \times 19}{2 \times 1} = 190 \)**<br /><br />Therefore, the correct answer is (D) 190.