Permutations And Combinations

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 items from a set of distinct items (where order doesn't matter) is given by the combination formula:
5. Apply to the Problem:
   • Total number of people, .
   • Number of people involved in each handshake, .
   • We need to calculate the number of combinations of choosing 2 people from 20, which is or .
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 .

The correct option is (B) 20C2.