Number of handshakes when every person in a group of 23 people shakes hands with every other person exactly once

Introduction / Context:
This is a classic handshake problem that tests your understanding of combinations. Each handshake involves a pair of distinct people, and every pair shakes hands exactly once. You must avoid double counting pairs.

Given Data / Assumptions:

• There are 23 people in a group.
• Each pair of distinct people shakes hands exactly once.
• No person shakes hands with themselves.
• The order inside a pair does not matter, so person A shaking hands with person B is the same as person B shaking hands with person A.

Concept / Approach:
Each handshake corresponds to a unique unordered pair of people. When there are n people, the number of unordered pairs is given by the combination nC2. This is because we must choose 2 distinct people out of n without considering order.

Step-by-Step Solution:
Step 1: Let n = 23, the total number of people.Step 2: The number of distinct handshakes equals the number of ways to choose 2 people from 23.Step 3: Use the formula for combinations: nC2 = n * (n - 1) / 2.Step 4: Substitute n = 23 to get 23C2 = 23 * 22 / 2.Step 5: Compute 23 * 22 = 506 and then divide by 2 to get 506 / 2 = 253.Step 6: Therefore, total handshakes = 253.

Verification / Alternative check:
Another way to see this is to imagine that each of the 23 people shakes hands with the remaining 22 people. This gives 23 * 22 handshake actions, but each handshake is counted twice (once from each person), so you divide by 2. That again gives 23 * 22 / 2 = 253.

Why Other Options Are Wrong:

• 142, 175, and 212 are smaller than 253 and do not correspond to the correct combination computation.
• They may result from arithmetic mistakes or attempts to subtract extra counts incorrectly.

Common Pitfalls:
Some learners mistakenly use permutations and compute 23P2, which counts ordered pairs and gives a much larger number. Others might forget to divide by 2 when using the product method. Always remember that each handshake is an unordered pair and that nC2 is the correct formula.

Final Answer:
The total number of handshakes in the group is 253, so the correct answer is 253.