3.4. Explain in words the number of handshakes 3.5. Write a formula whicl participants and the nun

It seems like your question is about calculating the number of handshakes in a scenario where a certain number of participants are involved. This is a common problem in combinatorics, often referred to as the "handshake problem." ### 3.4. Explanation of the Number of Handshakes When each participant shakes hands with every other participant exactly once, the total number of handshakes can be understood as follows: - If there are \( n \) participants, each participant can potentially shake hands with \( n - 1 \) others. - However, if we count this way, we will count each handshake twice (once for each participant involved in the handshake). Therefore, we need to divide the total by 2 to avoid double counting. For example, if there are 4 participants (A, B, C, D): - A shakes hands with B, C, and D (3 handshakes). - B shakes hands with C and D (2 additional handshakes, but A-B is already counted). - C shakes hands with D (1 additional handshake, but A-C and A-D are already counted). Thus, the total number of unique handshakes is 6, which can be calculated as: - A-B, A-C, A-D, B-C, B-D, C-D. ### 3.5. Formula for the Number of Handshakes The formula to calculate the number of unique handshakes among \( n \) participants is given by: \[ H = \frac{n(n - 1)}{2} \] Where: - \( H \) is the total number of handshakes. - \( n \) is the number of participants. This formula arises from the fact that we are choosing 2 participants from \( n \) to form a handshake, which is a combination problem. The number of ways to choose 2 participants from \( n \) is represented mathematically as \( \binom{n}{2} \), which simplifies to \( \frac{n(n - 1)}{2} \). ### Summary In summary, the number of handshakes among \( n \) participants can be calculated using the formula \( H = \frac{n(n - 1)}{2} \), which accounts for the fact that each handshake is counted twice when considering each participant's perspective.