Question

Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place?

• A. \( ^{19}C_2 \)
• B. \( ^{20}C_2 \)
• C. \( ^{20}C_{19} \)
• D. \( 20^2 \)

Hint:

The number of ways to choose 2 people from \( n \) for a handshake is \( \binom{n}{2} = \frac{n(n-1)}{2} \).

Answer 1

The Correct Option is B

1. Understand the Problem: We have a group of 20 people, and we need to find the total number of handshakes if every person shakes hands with every other person exactly once.
2. Nature of Handshakes: A handshake involves exactly two people. The order doesn't matter (Person A shaking hands with Person B is the same handshake as Person B shaking hands with Person A).
3. Relate to Combinations: Since the order doesn't matter and each handshake involves selecting a pair of people, this is a problem of finding the number of ways to choose 2 people from a group of 20. This is a combination problem.
4. Combination Formula: The number of ways to choose a subset of \(k\) items from a set of \(n\) distinct items (where order doesn't matter) is given by the combination formula: \[ \binom{n}{k} = nCk = \frac{n!}{k!(n-k)!} \]
5. Apply to the Problem:
   • Total number of people, \(n = 20\).
   • Number of people involved in each handshake, \(k = 2\).
   • We need to calculate the number of combinations of choosing 2 people from 20, which is \(\binom{20}{2}\) or \(20C2\).
6. Compare with Options:
   • (A) 19C2: Represents choosing 2 from 19 people.
   • (B) 20C2: Represents choosing 2 from 20 people. This matches our reasoning.
   • (C) 20C19: Represents choosing 19 from 20 people (which equals 20C1 = 20).
   • (D) 202: This doesn't represent the combination calculation.

The number of handshakes is the number of ways to choose 2 people from 20, which is represented by \(20C2\).

The correct option is (B) 20C2.

Answer 2

Total number of handshakes is given by: \[ \binom{20}{2} = \frac{20 \times 19}{2} = 190 \] Conclusion: The total number of handshakes is 190.