Eight people enter a lounge and each pair of people shake hands exactly once. What is the total number of handshakes?

Introduction / Context:
Handshakes between pairs correspond to choosing 2 people from the group. This is the classic handshake (graph edges) problem.

Given Data / Assumptions:

• n = 8 distinct people.
• Each handshake involves a unique pair; no repeats, no self-handshakes.

Concept / Approach:
The number of unique pairs from n is C(n, 2) = n(n − 1)/2.

Step-by-Step Solution:

Verification / Alternative check:
Interpret as edges in the complete graph K8; edges = C(8, 2) = 28.

Why Other Options Are Wrong:
16, 36, and 56 are typical distractors; only 28 equals C(8, 2).

Common Pitfalls:
Double-counting ordered pairs (8P2 / 2! is the correct unordered count).

Final Answer:
28