Question

A team of 11 players is to be chosen from a group of 15 candidates. (a) How many different teams can be chosen? (b) How many teams can be chosen if one player is designated captain and must play on the team?

Short answer

(a) 1365 teams (b) 1001 teams

Step-by-step solution

Understand the Combination Formula

To solve problems involving combinations, we use the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \(!\) denotes factorial. This calculates the number of ways to choose \( r \) items from \( n \) without regard to the order of selection.

Calculate Total Number of Teams (Part a)

We need to find the number of ways to choose 11 players out of 15. Using the combination formula: \( \binom{15}{11} = \binom{15}{4} \). This simplifies the calculation due to the symmetry property of combinations: \( \binom{n}{r} = \binom{n}{n-r} \). Calculate \( \binom{15}{4} = \frac{15!}{4! \times 11!} \).

Simplify the Calculation

Calculate the factorial of the numbers involved: \( 15! = 15 \times 14 \times 13 \times 12 \times 11! \). Hence, \( \frac{15!}{11!} = 15 \times 14 \times 13 \times 12 \). Then divide by \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Finally, \( \frac{15 \times 14 \times 13 \times 12}{24} = 1365 \).

Identify Constraints for Part b

For part (b), one player must be on the team as the captain. Essentially, we are choosing the remaining 10 players from the other 14 candidates. Thus, we use the formula \( \binom{14}{10} = \binom{14}{4} \).

Simplify the Calculation for Part b

Use the formula \( \binom{14}{4} = \frac{14!}{4! \times 10!} \). Simplify this as \( \frac{14 \times 13 \times 12 \times 11}{24} = 1001 \).

Conclude the Calculations

The final number of teams in part (a) is 1365. For part (b), where one specific player is guaranteed to be on the team, there are 1001 possible selections for the rest of the team.

Key concepts

Combination formula

The combination formula is a crucial tool in combinatorics that allows us to calculate the number of ways to choose a specific number of items from a larger group without considering the order. This mathematical formula is expressed as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). In this formula:

• \( n \) is the total number of items available.
• \( r \) is the number of items to be chosen.
• \(!\) denotes a factorial, which we will discuss in the next section.

Notably, the combination formula emphasizes that the order of selection does not matter. This aspect differentiates combinations from permutations, where order does matter. In our problem, we're interested in forming teams of players, where the sequence of picking them is irrelevant. The combination formula helps us determine how many groups of players are possible. This is why it is perfect for determining all possible teams that can be created from a large group of candidates.

Factorial

Factorials are a fundamental part of calculations in combinatorics, especially when using the combination formula. When we mention \( n! \) in mathematics, we're referring to the factorial of \( n \). This is calculated by multiplying all positive integers less than or equal to \( n \). For example, the factorial for 4, written as \( 4! \), is calculated as:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorials grow rapidly, which can make calculations quite large but are simplified in combinatorics because many terms often cancel out. For instance, when calculating \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), the factorial terms in the denominator help simplify the seemingly huge factorial products that appear in \( n! \). Understanding how to work with factorials, simplifying equations through cancelation, is key in solving combinatorial problems efficiently.

Problem solving

Problem-solving in combinatorics often involves using a methodical approach, typically conducted through a series of well-defined steps. When faced with a problem, such as choosing teams from a set of candidates, employing combinations and factorial concepts is essential.
First, clearly define the problem and identify the constraints, like the number of items you need to choose and understand whether the order of selection matters. Then, use the correct formula to set up the equation based on the given data. In our example, for part (a), we calculate the combinations of choosing 11 from 15, while for part (b), we must also account for a pre-decided player, recalculating for 10 players from 14.

Breaking down the formula and plugging in the numbers involves understanding math operations like factorial simplifications, which make the problem manageable. Each step in the process builds towards the solution, reinforcing the application of mathematical theory to practical scenarios. This structured process not only helps in resolving specific exercises but also enhances skills for tackling various complex problems in future endeavors.