Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

A football squad consists of three centers, ten linemen who can play either guard or tackle, three quarterbacks, six halfbacks, four ends, and four fullbacks. A team must have one center, two guards, two tackles, two ends, two halfbacks, a quarterback, and a fullback. In how many different ways can a team be selected from the squad?

Step-by-Step Solution

Answer

4,082,400 ways to form the team.

Step 1: Select the Center

There are 3 centers to choose from for the position of the center in the team. So, the number of ways to choose a center is 3.

Step 2: Select the Guards

We have 10 linemen who can play either guard or tackle. We need to choose 2 to fill the guard positions. This can be done in \( \binom{10}{2} \) ways, where \( \binom{n}{k} \) is the combination formula \( \frac{n!}{k!(n-k)!} \). Calculating we get \( \binom{10}{2} = 45 \).

Step 3: Select the Tackles

After selecting 2 players as guards, 8 linemen remain. Now, we need to choose 2 for the tackle positions. This can be done in \( \binom{8}{2} \) ways. Calculating we get \( \binom{8}{2} = 28 \).

Step 4: Select the Ends

There are 4 ends. We need to select 2 ends for the team. This can be done in \( \binom{4}{2} \) ways. Calculating gives \( \binom{4}{2} = 6 \).

Step 5: Select the Halfbacks

There are 6 halfbacks available, and we need to select 2 for the team. This can be done in \( \binom{6}{2} \) ways. Calculating gives \( \binom{6}{2} = 15 \).

Step 6: Select the Quarterback

There are 3 quarterbacks available, and we need to choose 1 for the team. So, there are 3 ways to select the quarterback.

Step 7: Select the Fullback

There are 4 fullbacks, and we need to choose 1 for the team. So, there are 4 ways to select the fullback.

Step 8: Calculate the Total Number of Teams

To find the total number of ways to form the team, we multiply the number of ways to choose each position: \[ 3 \times 45 \times 28 \times 6 \times 15 \times 3 \times 4 \]Calculate the product to get the total ways.\[ 3 \times 45 = 135 \]\[ 135 \times 28 = 3780 \]\[ 3780 \times 6 = 22680 \]\[ 22680 \times 15 = 340200 \]\[ 340200 \times 3 = 1020600 \]\[ 1020600 \times 4 = 4082400 \]Thus, the total number of ways to form the team is 4,082,400.

Key Concepts

Combination Formula

In combinatorics, the combination formula is an essential tool. It's used to determine how many ways you can choose a specific number of items from a larger group, without regard to the order of selection. This is particularly useful in situations like team selection where the order doesn't matter, only the combination of players does.
The combination formula is given by:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \( n \) is the total number of items to choose from, \( k \) is the number of items to choose, and \(!\) denotes factorial, which is the product of all positive integers up to that number.
In the given exercise, this formula helps to determine how many ways you can choose, for example, 2 guards out of 10 linemen. Using the formula,

• \( \binom{10}{2} = \frac{10!}{2!(8!)} = 45 \)

This tells us that there are 45 possible ways to choose 2 guards from 10 linemen.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics. Understanding the difference between the two is crucial for solving mathematical problems related to arrangements and selections.
Permutations involve the arrangement of items where order does matter. For example, different ways to arrange players in different positions where each position means something unique.
Combinations, on the other hand, involve selection without regard to order. In the context of team selection, we only care about which players are chosen, not the order in which they are picked. This is why combinations are used in this problem. We use combinations to select players for each position group where the order is irrelevant.
When selecting guards, ends, halfbacks, or tackles, the focus is only on who gets selected, allowing us to employ the combination formula efficiently.

Team Selection

Team selection is a common problem-solving scenario in combinatorics. This exercise gives a practical example of choosing a subset of players from a larger pool to form a team. This process involves selecting players for different roles such as center, guards, tackles, quarterbacks, ends, halfbacks, and fullbacks.
Each role requires a specific number of players, and we use combinatorial methods to determine how many different teams can be created. In the football squad example:

• 1 center is chosen from 3 options.
• 2 guards are selected from 10 linemen.
• 2 tackles from the remaining linemen.
• 2 ends from 4 players.
• 2 halfbacks from 6 available players.
• 1 quarterback from 3 candidates.
• 1 fullback from 4 choices.

Using combinations for these selections ensures all possible teams are accounted for without considering the order of selection within each position group.

Mathematical Problem Solving

Mathematical problem solving involves a systematic approach to understand and solve a problem. In combinatorics, this involves breaking down a problem into manageable pieces using established formulas and principles.
The exercise of team selection from a football squad is an example where breaking it down step by step is beneficial. By calculating the number of ways to choose players for each position, and then multiplying these results, we find the total number of team combinations. These steps include:

• Identifying how many players are needed for each position.
• Applying the combination formula to calculate the number of ways to select those players.
• Multiplying these numbers to find the total number of ways to form the team.

It's a clear and logical process that demonstrates how powerful mathematical techniques are for solving real-world selection problems.