Question
Ten people are in a room. Each person shakes hands with every other person in the room exactly once.
How many handshakes will there be?
How many handshakes will there be?
Answers
GPT-4o mini
To determine the total number of handshakes among ten people where each person shakes hands with every other person exactly once, we can use the formula for combinations.
The formula for the number of ways to choose 2 people from \( n \) people is given by:
\[
\binom{n}{2} = \frac{n(n-1)}{2}
\]
In this case, \( n = 10 \). Plugging this into the formula, we get:
\[
\binom{10}{2} = \frac{10 \cdot 9}{2} = \frac{90}{2} = 45
\]
Thus, the total number of handshakes is \( 45 \).
Therefore, the answer is:
\[
\boxed{45}
\]
The formula for the number of ways to choose 2 people from \( n \) people is given by:
\[
\binom{n}{2} = \frac{n(n-1)}{2}
\]
In this case, \( n = 10 \). Plugging this into the formula, we get:
\[
\binom{10}{2} = \frac{10 \cdot 9}{2} = \frac{90}{2} = 45
\]
Thus, the total number of handshakes is \( 45 \).
Therefore, the answer is:
\[
\boxed{45}
\]