Solving the Handshake Problem: An In-Depth Analysis

The handshake problem is a classic combinatorial mathematics question often used to test basic understanding of combinatorial principles. In this article, we will delve into how to solve the handshake problem by applying combinatorial mathematics and factorizing quadratic equations. Consider the following: if a total of 190 distinct handshakes was made at a party, how many people were present?

The Problem Statement

At a school or private gathering, every person shakes hands with every other person exactly once. Given a total of 25 handshakes in a particular gathering, we aim to determine how many people attended. The calculation involves understanding that each handshake involves two people. Therefore, for X people at the gathering, the total number of handshakes is given by (X-1)/2.

Calculations for Smaller Groups

For 6 people, the number of handshakes is calculated as follows:

6(6-1)/2 15 handshakes

For 7 people, the calculation is:

7(7-1)/2 21 handshakes

For 8 people, the calculation results in:

8(8-1)/2 28 handshakes

For 9 people, the total is:

9(9-1)/2 36 handshakes

For 10 people, the handshake count is:

10(10-1)/2 45 handshakes

Understanding the Quadratic Equation

We are given that the total number of handshakes is 190. Thus, we can set up the equation for the number of people as follows:

n(n-1)/2  190

Multiplying both sides by 2 to clear the fraction, we get:

n2 - n - 380  0

Solving the Quadratic Equation

This is a quadratic equation in the standard form ax2 bx c 0, where a 1, b -1, and c -380. The quadratic formula to solve for N is given by:

N  [-b ± √(b2 - 4ac)] / (2a)

Substituting the values, we get:

N  [1 ± √(1   1520)] / 2    [1 ± √1521] / 2    [1 ± 39] / 2

Evaluating the Solutions

From the solutions obtained, we discard the negative value as the number of people cannot be negative. Therefore, the number of people is:

N  (1   39) / 2  20

Hence, there are 20 people at the party.

Further Exploration

The choose notation, denoted nC2, is also applicable to solve such problems. It is defined as:

nC2  n(n-1)/2

This formula represents the number of ways to choose 2 items from n items, a fundamental concept in combinatorial mathematics.

In summary, the key to solving handshake problems lies in understanding combinatorial principles and utilizing quadratic equations to find the solution. Whether you have a small group or a larger party, the underlying mathematics remains consistent.

Related Keywords

Handshakes Party math Combinatorial mathematics

1. The solutions to the equation n2 - n - 380 0 are N 20 and N -19. Since the number of people must be a positive integer, we discard -19.