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?
Answer options with 5 options
A.
90
B.
45
C.
100
D.
19
E.
55
How many handshakes will there be?
Answer options with 5 options
A.
90
B.
45
C.
100
D.
19
E.
55
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. Specifically, we want to find the number of ways to choose 2 people from a group of 10 (since a handshake involves 2 people).
The formula for combinations is given by:
\[
C(n, k) = \frac{n!}{k!(n-k)!}
\]
where \(n\) is the total number of items (people) and \(k\) is the number of items to choose. In this case, \(n = 10\) and \(k = 2\):
\[
C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!}
\]
Calculating this gives:
\[
C(10, 2) = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45
\]
Thus, the total number of handshakes is 45.
The answer is **B. 45**.
The formula for combinations is given by:
\[
C(n, k) = \frac{n!}{k!(n-k)!}
\]
where \(n\) is the total number of items (people) and \(k\) is the number of items to choose. In this case, \(n = 10\) and \(k = 2\):
\[
C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!}
\]
Calculating this gives:
\[
C(10, 2) = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45
\]
Thus, the total number of handshakes is 45.
The answer is **B. 45**.