Question

In a group of 5 people, each shakes hands with every other person exactly once. How many handshakes occur?

• A. 10
  
• B. 12
  
• C. 15
  
• D. 20
  

Hint:

For handshakes, use $\binom{n}{2}$ to count pairs efficiently.

Answer 1

The Correct Option is A

- Step 1: Understand the problem. Each of 5 people shakes hands with 4 others, but each handshake involves 2 people.
- Step 2: Use combinations. Number of handshakes = number of ways to choose 2 people: $\binom{5}{2}$.
- Step 3: Calculate. $\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$.
- Step 4: Alternative approach. Each person shakes hands with 4 others: $5 \times 4 = 20$. Divide by 2 (each handshake counted twice): $\frac{20}{2} = 10$.
- Step 5: Compare with options. Options: (1) 10, (2) 12, (3) 15, (4) 20. Matches 10.
- Step 6: Verify. For 5 people (A, B, C, D, E), handshakes are AB, AC, AD, AE, BC, BD, BE, CD, CE, DE = 10.
- Step 7: Conclusion. Option (1) is correct.