Problem 69 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Hockey Lineup A hockey team has 20 players, of whom 12 play forward, six play defense, and two are goalies. In how many ways can the coach pick a starting lineup consisting of three forwards, two defense players, and one goalie?

Step-by-Step Solution

Answer

There are 6600 ways to pick the starting lineup.

Step 1: Identify the Problem

We need to form a starting lineup consisting of 3 forwards, 2 defense players, and 1 goalie from a team of hockey players.

Step 2: Use the Combination Formula for Forwards

Calculate the number of ways to choose 3 forwards out of 12. The formula for combinations is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Substituting in the values: \( \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!} \). Calculating, we get \( \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \).

Step 3: Use the Combination Formula for Defense

Now calculate the number of ways to choose 2 defense players from 6. Using the combination formula: \( \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} \). Calculating, we find \( \frac{6 \times 5}{2 \times 1} = 15 \).

Step 4: Use the Combination Formula for Goalies

Since we need to choose 1 goalie from 2, use the formula again: \( \binom{2}{1} = \frac{2!}{1!(2-1)!} = \frac{2}{1} = 2 \).

Step 5: Calculate Total Ways Using Multiplication Principle

To find the total number of ways to form the lineup, multiply the results: \( 220 \times 15 \times 2 = 6600 \). This product represents the total number of ways to choose and organize the players into the specified lineup.

Key Concepts

Combination Formula

In the realm of combinatorics, the combination formula is essential for calculating the number of ways to choose a subset of items from a larger set, without considering the order of selection. This formula is particularly useful for problems where you need to select items and the order doesn't matter, as was the case in our hockey lineup problem.

The formula is typically expressed as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where

• \( n \) is the total number of items to choose from,
• \( k \) is the number of items to be chosen, and
• \( ! \) denotes factorial, which is the product of all positive integers up to that number.

For example, to calculate how many ways to choose 3 forwards from 12, we use the formula to find \( \binom{12}{3} = 220 \).

This approach was similarly applied to select defense players and goalies, reinforcing the power of the combination formula in simplifying complex selection problems.

Counting Principles

Counting principles form the foundation of combinatorics, guiding us on how to systematically count outcomes without error or omission. They allow us to compute the total number of ways an event can occur. There are two main techniques: the addition principle and the multiplication principle.

In the hockey lineup problem, the addition principle is not directly applicable since it applies when events are mutually exclusive, meaning only one event can occur. Instead, we focus on the multiplication principle, where the events occur in sequence, such as choosing different positions in the lineup.

The counting principles help ensure that all possible lineup combinations are accounted for, ensuring a comprehensive approach to solving such combinatorial problems.

Multiplication Principle

The multiplication principle is a key counting technique used for finding the number of possible outcomes in multi-stage processes. It states that if one event can occur in \( m \) ways and a second event can occur independently of the first in \( n \) ways, then both events can occur in sequence in \( m \times n \) ways.

In our problem, the starting lineup consists of three stages:

• Choosing 3 forwards from 12 available players
• Picking 2 defense players from 6 available
• Selecting 1 goalie from 2

To find the total number of ways these events can occur, we multiply the number of possibilities from each stage: \[ 220 \times 15 \times 2 = 6600\]Thus, utilizing the multiplication principle effectively combines the selections of forwards, defense players, and goalies into a single comprehensive calculation.

Hockey Lineup Problem

The hockey lineup problem is a classic example of applying combinatorial counting techniques to real-world scenarios. The challenge is to determine the number of ways to arrange a distinct group of players into a lineup, based on specified criteria or constraints.

In this case, the task was to form a lineup of 3 forwards, 2 defense players, and 1 goalie from the given team composition. By breaking down the problem step-by-step, we use:

• The combination formula to choose players for each position
• The multiplication principle to calculate the total number of lineups

The result is a total of 6600 unique ways to assemble the starting lineup.

This problem exemplifies how understanding foundational combinatoric principles can simplify and solve complex selection tasks effectively.