Question

At the end of a business conference, the ten people present all shake hands with each other once. How many handshakes will there be altogether?

• A. 20
• B. 45
• C. 55
• D. 90
• E.

Hint:

In handshake problems, use the combination formula to determine the total number of possible handshakes.

Answer 1

The Correct Option is B

For each handshake, two people shake hands. The total number of handshakes can be found using the formula for combinations \( C(n, 2) \), where \( n \) is the number of people: \[ C(10, 2) = \frac{10 \times 9}{2} = 45. \] Thus, the total number of handshakes is 45. So, the correct answer is (b) 45. 

Answer 2

Instead of using the combination formula directly, count the handshakes person by person as each new person is introduced.

1. The first person shakes hands with the other 9 people: 9 handshakes.
2. The second person has already shaken hands with the first, so only shakes hands with the remaining 8 new people: 8 handshakes.
3. Continuing this way, the third person adds 7 new handshakes, the fourth adds 6, and so on, down to the ninth person adding 1 new handshake with the tenth.
4. Adding these up: \( 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 \).

This matches the total obtained from the combination approach, confirming the count.

Hence, the correct answer is 45.