Question

There are 20 people in a party. If every person shakes hand with every other person, what is the total number of handshakes?

• A. 180
• B. 155
• C. 145
• D. 190

Hint:

For any "handshake" or "round-robin tournament match" problem with \(n\) participants, always use the formula \(\frac{n(n-1)}{2}\) to find the total combinations quickly.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A handshake between two people requires exactly a combination of two individuals chosen from the total group.
Since the order in which the handshake occurs does not matter (person A shaking hands with person B is the same as person B shaking hands with person A), we use the mathematical concept of combinations.
Key Formula or Approach:
The number of ways to choose \(r\) objects from a set of \(n\) distinct objects is given by the combination formula:
\[ ^nC_r = \frac{n!}{r!(n-r)!} \]
For a handshake between 2 people out of a total of \(n\) people, the formula simplifies to:
\[ \text{Total Handshakes} = \frac{n(n-1)}{2} \]

Step 2: Detailed Explanation:
Given the total number of people in the party is \(n = 20\).
We need to calculate the total number of unique handshakes:
\[ \text{Total Handshakes} = \frac{20 \times (20 - 1)}{2} \]
\[ \text{Total Handshakes} = \frac{20 \times 19}{2} \]
\[ \text{Total Handshakes} = 10 \times 19 = 190 \]
Thus, if every person shakes hands with every other person exactly once, there will be a total of 190 handshakes.

Step 3: Final Answer:
The total number of handshakes is 190 (Option D).