Twenty persons shake hands with one another at a party. How many shake hands took place?
20
200
120
190

To determine the number of handshakes that took place, we can use the combination formula. The formula for calculating the number of ways to choose 2 people out of \( n \) people (which corresponds to a handshake between two people) is given by:

\[
\text{Number of handshakes} = \binom{n}{2} = \frac{n(n-1)}{2}
\]

Here, \( n = 20 \):

\[
\text{Number of handshakes} = \frac{20 \times 19}{2} = \frac{380}{2} = 190
\]

So, the number of handshakes that took place is **190**.

Would you like more details or have any questions? 

Here are 5 related questions to expand on this topic:

1. How would the number of handshakes change if there were 25 people instead of 20?
2. What is the general formula to calculate the number of handshakes for any number of people?
3. If each handshake took 5 seconds, how much total time was spent on handshakes?
4. How would you calculate the number of handshakes if two people refused to shake hands with each other?
5. How does this problem relate to graph theory?

**Tip:** For any problem involving counting pairs or combinations, the combination formula \(\binom{n}{2}\) is very useful.

Learn how to calculate the number of handshakes among 20 people at a party using the combination formula. This problem involves understanding combinatorics and combinations, providing a fundamental insight into counting pairs. Suitable for Grades 7-9.

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