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Chapter 1: Problem 13

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

Short Answer

Expert verified
In a group of 20 people, there are \(C(20, 2) = 190\) unique handshakes that can take place, found using the combination formula.

Step by step solution

01

Understand the Problem

: We have a group of 20 people and want to find out how many unique handshakes can take place between them. Each handshake involves a pair of 2 individuals.
02

Apply the Combination Formula

: We will use the combination formula to find the number of handshakes. In this case, n = 20 (total people) and k = 2 (number of people in a handshake). The formula is given as: \(C(n, k) = \frac{n!}{k!(n-k)!}\).
03

Calculate Factorials

: Calculate the required factorials for the formula. We need the factorials of n (20), k (2), and (n-k) (20-2=18). \(20! = 20 \times 19 \times 18 \times ... \times 3 \times 2 \times 1\) \(2! = 2 \times 1\) \(18! = 18 \times 17 \times ... \times 3 \times 2 \times 1\)
04

Plug Factorials into the Formula

: Now that we have our factorials, plug them into the combination formula: \(C(20, 2) = \frac{20!}{2!(18)!}\)
05

Calculate the Number of Handshakes

: Evaluate the expression to find the number of unique handshakes between the 20 people: \(C(20, 2) = \frac{20 \times 19 \times 18 \times ... \times 3 \times 2 \times 1}{(2 \times 1)[18 \times 17 \times ... \times 3 \times 2 \times 1]}\) Observe that the terms 18! cancel out from the numerator and the denominator: \(C(20, 2) = \frac{20 \times 19}{2}\) Multiply and divide the expression: \(C(20, 2) = \frac{380}{2}\) \(C(20,2) = 190\) The number of unique handshakes between 20 people is 190.

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