23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?
Note that if each person shakes hands with every other person, then the number of handshakes is maximized. There are $\binom{23}{2} = \frac{(23)(22)}{2} = (23)(11) = 230+23 = \boxed{253}$ ways to choose two people to form a handshake.
253
Level 2
Counting & Probability
Problem: 23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the maximum number of handshakes at the party, we need to consider the nature of handshakes. Each handshake involves two distinct people. Therefore, if we have 23 people, we can calculate the total number of unique pairs of people who can shake hands.

The formula for the number of ways to choose 2 people from a group of n people is given by the combination formula C(n, 2), which is calculated as:

C(n, 2) = n! / (2!(n - 2)!)

For our case, n = 23. Thus, we can substitute 23 into the formula:

C(23, 2) = 23! / (2!(23 - 2)!) = 23! / (2! * 21!) 

This simplifies to:

C(23, 2) = (23 * 22) / (2 * 1) 

Calculating this gives:

C(23, 2) = 506 / 2 = 253

This calculation shows that the maximum number of handshakes that can occur, given that each person shakes hands with every other person exactly once, is 253. Therefore, the final answer is 253.<|END_OF_LATENT|> The final answer is $253$.