Problem 68 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Casting a Play A group of 22 aspiring thespians contains 10 men and 12 women. For the next play the director wants to choose a leading man, a leading lady, a supporting male role, a supporting female role, and eight extras- three women and five men. In how many ways can the cast be chosen?

Step-by-Step Solution

Answer

There are 274,909,800 ways to choose the cast.

Step 1: Choose the leading man

The director needs to choose 1 leading man from the 10 men available. The number of ways to do this is simply choosing 1 man from 10, calculated as \( \binom{10}{1} = 10 \).

Step 2: Choose the leading lady

The director needs to choose 1 leading lady from the 12 women available. This can be done in \( \binom{12}{1} = 12 \) ways.

Step 3: Choose the supporting male role

After choosing the leading man, 9 men remain for the supporting male role which can be chosen in \( \binom{9}{1} = 9 \) ways.

Step 4: Choose the supporting female role

After choosing the leading lady, 11 women remain for the supporting female role which can be chosen in \( \binom{11}{1} = 11 \) ways.

Step 5: Choose the female extras

For the role of female extras, 3 women need to be selected from the remaining 11 women. The number of combinations possible is \( \binom{11}{3} = 165 \) ways.

Step 6: Choose the male extras

For the role of male extras, 5 men need to be selected from the remaining 9 men. This can be done in \( \binom{9}{5} = 126 \) ways.

Step 7: Calculate the total number of ways

To find the total number of ways to choose the cast, multiply the number of ways to select each role: \(10 \times 12 \times 9 \times 11 \times 165 \times 126 = 274,909,800\).

Key Concepts

Counting Principles

Counting principles are the foundation of combinatorics, which deals with counting, arranging, and measuring different configurations. They help solve problems where you need to calculate the number of ways certain events can occur. Two fundamental counting principles play a key role: the Addition Principle and the Multiplication Principle.
The **Addition Principle** states that if there are multiple ways to do something, and these ways do not overlap, then you add the number of possibilities. For instance, if you can either choose a red shirt in 5 ways or a blue shirt in 3 ways, there are a total of 8 ways to choose a shirt.
On the other hand, the **Multiplication Principle** is used when tasks are done in sequence. For example, if you have 4 ways to select a hat and 3 ways to choose a pair of shoes, you multiply these choices to find that there are 12 possible combinations of hats and shoes you can wear. This principle is crucial in our theater casting problem, where we find solutions by multiplying the number of ways to select each member of the cast.
The beauty of counting principles lies in their simplicity and broad applications, as shown by the theater problem where each role must be filled in a step-by-step process.

Binomial Coefficients

Binomial coefficients, represented as \( \binom{n}{k} \), are numbers that tell you how many ways you can choose \( k \) items from a set of \( n \) items. This is pivotal in combinatorial mathematics and is often called "n choose k."
The formula for calculating the binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \( n! \) (n factorial) means you multiply all whole numbers from 1 up to \( n \). These coefficients find a spot in diverse areas from probability to algebra. They show up as entries in Pascal's Triangle, where each number is the sum of the two numbers directly above it.
In our theater casting problem, we use binomial coefficients to select roles where order does not matter. Choosing extras, for example, involves selecting sets of individuals from available pools (e.g., 3 women from a remaining pool of 11). Thus, using binomial coefficients helps determine the number of possible groups you can form, allowing us to calculate different arrangements efficiently.

Theater Casting Problem

Our theater casting problem is an interesting real-world application of combinatorics. The aim is to distribute specific roles among a group of actors based on set conditions. This challenge tests the understanding of counting principles and binomial coefficients.
Each role from the leading parts to the extras requires us to decide who will fill them. For leads, since each is a distinct role, the subtraction of each chosen actor from future selections (e.g., picking 1 leading man leaves 9 for a supporting role) is important. The binomial coefficients allow us to handle roles where groups are necessary, such as selecting 5 male extras out of 9 remaining candidates.
Such problems need careful reading to pick apart requirement details: how many people fit each role, and how the selections affect subsequent choices. By multiplying the possibilities of each choice (man by man, woman by woman), we compute essential figures in diverse configurations of the cast.

• Choose a leading man or woman: simple selection leads to direct subtraction from the pool.
• Decide on extras: requires determining combinations of remaining choices.
• Integration of both decisions through multiplication yields the complete set of solutions.

This framework not only clarifies casting but strengthens core combinatorial skills applicable in various decision-making scenarios.