Problem 42 - Thinking Mathematically

Question

There are five people in a room. Each person shakes the hand of every other person exactly once. How many handshakes are exchanged?

Step-by-Step Solution

Answer

The total number of handshakes exchanged is 10.

Step 1: Understand the nature of the event

The first step is realizing that when a person shakes hands with another, it involves two people. When a third person shakes hands, this event does not depend on the first two people shaking hands. This means these are combinations, not permutations. Hence the formula \(C(n, k) = n! / [k!(n-k)!]\) will be useful here.

Step 2: Identify what 'n' and 'k' are

Here, the total number of people 'n' is 5 and the number of people taking part in one handshake 'k' is 2.

Step 3: Calculate the combination

Substitute 'n' and 'k' into the combination formula which results in \(C(5, 2) = 5! / [2!(5-2)!]\)

Step 4: Calculate the Factorials

Calculating the factorials, we have \(5! = 5 × 4 × 3 × 2 × 1 = 120\), \(2! = 2 × 1 = 2\), and \((5-2)! = 3! = 3 × 2 × 1 = 6\)

Step 5: Substitute back into the formula

Substituting back into the formula, the result becomes \(C(5, 2) = 120 / (2 × 6) = 10\).

Key Concepts

Understanding Combinations

Combinations are a fundamental concept in combinatorics, helping to determine the number of ways to select items from a group without regard to order. Unlike permutations where order does matter, combinations focus solely on the selection. The formula for combinations is expressed as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here,

• \( n \) is the total number of items.
• \( k \) represents how many items are being chosen.

For example, when calculating how many ways handshakes can occur between five people, we choose two at a time because a handshake involves two people. Therefore, in the context of combinations, the order doesn't affect the handshake, just who shakes hands with whom.

The Role of Factorials

Factorials play a significant role in calculating combinations as well as in various other mathematics problems. A factorial, denoted by an exclamation mark \( n! \), is the product of all positive integers up to \( n \). It's crucial in combinations because it helps calculate the number of potential groupings. Here's how factorials are used: When the problem asks to find out handshakes among five people, you compute \( 5! \) to figure out the total permutations, but since order doesn’t matter here, you adjust the count by considering \( 2! \) for the pair and \( (5-2)! \) for the disregarded order. That's:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

This calculation ensures that your total reflection of possibilities (combinations) acknowledges the absence of order.

The Handshake Problem

The handshake problem is a delightful example of applying combinations to solve real-world-like problems. It involves determining the ways in which each person in a group can make contact with another, typically via a handshake, without repeating any interactions. Consider a room with 5 people: each person shakes hands with all others. The question is, "How many distinct handshakes occur?" Applying what we know:

• Use the combination formula to ensure you're counting pairs: \( C(5, 2) \).
• Substitute values: \( C(5, 2) = \frac{5!}{2!3!} \) leading to: \( \frac{120}{2 \times 6} = 10 \).

This means there are 10 unique handshakes, illustrating efficiently how a seemingly tricky problem can be broken down through combinations and factorials.