Problem 64 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Hockey Lineup A hockey team has 20 players of which twelve 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

The coach can pick the starting lineup in 6600 different ways.

Step 1: Total Forward Choices

Determine how many ways to choose 3 forwards out of the 12 available players. This is a combination problem, so we use the combinations formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]substitute to obtain \[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!} = 220 \] Thus, there are 220 different ways to choose 3 forwards.

Step 2: Total Defense Choices

Determine how many ways to choose 2 defense players from the 6 available. Again, apply the combinations formula: \[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = 15 \]So there are 15 different ways to choose 2 defense players.

Step 3: Total Goalie Choices

Determine how many ways to choose 1 goalie from the 2 available players. Using the combinations formula: \[ \binom{2}{1} = \frac{2!}{1!(2-1)!} = \frac{2!}{1!1!} = 2 \]Thus, there are 2 different ways to choose 1 goalie.

Step 4: Calculate Total Lineups

The total number of lineups is the product of the number of ways to choose forwards, defense players, and goalie, because each group is chosen independently: \[ 220 \times 15 \times 2 = 6600 \]This multiplication gives us the total number of different starting lineups possible.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. This is useful in scenarios such as forming teams or picking lottery numbers, where you care only about the players selected and not the sequence.
Here's how combinations work:

• You have a total number of items, say \( n \).
• You need to choose a certain number, \( r \), from those items.
• The formula for combinations is: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In our hockey lineup problem, we use combinations to determine how many ways there are to choose forwards, defense players, and a goalie. For example, choosing 3 forwards from 12 involves calculating \( \binom{12}{3} \), which equals 220. This represents all possible groups of 3 forwards that can be picked from 12, irrespective of the order in which they are chosen.

Permutations

Permutations are similar to combinations, but with one key difference: order matters. Think of permutations like arranging books on a shelf, where the sequence they are placed in is important.
For permutations, the formula is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Here, \( n \) is the total number of items and \( r \) is how many you are choosing.
In our hockey problem, if we were concerned with the order in which the players were selected, we would use permutations. However, since each group selection is independent (e.g., choosing 3 forwards), we're only interested in combinations. But, understanding permutations helps differentiate situations where order impacts outcomes.

Factorial

Factorials are the product of all positive integers up to a certain number. It's a fundamental concept in permutations and combinations.
The notation for factorial is: \( n! \), which means:

• \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \)
• For example: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Using factorials simplifies computations in combination and permutation formulas. In the hockey problem, factorials help calculate how many different lineups are possible through combinations. By using \( 12! \), \( 3! \), and \( 9! \), we simplify finding the number of ways to choose 3 forwards. Factorials make dealing with large numbers manageable and are key to solving these types of problems efficiently.

Probability

Probability is the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain). When we consider arrangements and selections, probability helps determine the chances of specific combinations or permutations.
Probability basics:

• Probability of an event \( A \): \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

While the hockey problem doesn't directly ask for probability, understanding combinations helps you see the total number of lineups and pick out specific ones. If the coach wanted to find out the probability of a particular lineup, they would compare the number of favorable lineups to the 6600 total possibilities. Probability helps evaluate how likely an event is to happen based on these combinations and permutations.