Problem 4 - Thinking Mathematically

Question

In how many ways can a casting director choose a female lead and a male lead from five female actors and six male actors?

Step-by-Step Solution

Answer

The director can choose the female and male leads in 30 different ways.

Step 1: Identify Values

The problem can be divided into two parts. At first, the casting director has to select one female lead out of five female actors. In the next decision, the casting director has to select one male lead out of six male actors. Thus, first part involves 5 female actors available with only 1 required, and second part involves 6 male actors available with only 1 required.

Step 2: Use Combination formula for Female Actors

We use the combination formula to determine the number of ways to select one female actor from a group of five. Using the combination formula \( C(5, 1) = \frac{5!}{1!(5-1)!} \) gives us 5 ways.

Step 3: Use Combination formula for Male Actors

Next, we use the same formula to determine the number of ways to select one male actor from a group of six. Using the combination formula \( C(6, 1) = \frac{6!}{1!(6-1)!} \) gives us 6 ways.

Step 4: Combine the Results

As these are independent decisions, we can multiply the outcomes to get the overall number of ways to select a female and a male lead. Thus the result \( 5 * 6 \) gives us 30 ways.

Key Concepts

Combination Formula

Selecting a specific group from a larger set can be done using the combination formula. This is useful when the order of selection doesn't matter, which is different from permutations. The combination formula is expressed as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Here, \( n \) represents the total number of items to choose from, and \( r \) is the number of items to select. The "\(!\)" symbol called factorial, is the product of all positive integers up to a certain number.

In our casting scenario, selecting one actor is like finding combinations where we simply choose "1" from the total list either male or female. Thus, for the female lead, it's \( C(5, 1) \) which resolves to 5 ways. Similarly for the male actors, \( C(6, 1) \) resolves to 6 ways.

Independent Events

Independent events are situations where the occurrence of one event does not affect the occurrence of another. This concept is essential in probability calculations where different choices or events do not influence each other.

In the casting example, choosing the female lead is independent of choosing the male lead. Picking one does not change the options or outcome of the other selection. This means that for both the female and male actors, the number of ways to choose remains unaffected by the other gender's selection.

• Since both events are independent, you can calculate the total number of ways by simply multiplying their individual possibilities.

Probability Calculations

Probability calculations help us figure out the chances or likelihood of events happening. When dealing with multiple events, especially independent ones, these calculations often involve multiplying the probabilities of individual events.

For the casting problem, we need to calculate the number of ways the actors can be chosen, which is directly tied to the overall probability of any particular choice. The principle here is to multiply the outcomes of two independent events. For instance:

• You have 5 ways to choose the female lead and 6 ways to choose the male lead.
• Thus, the total number of ways to make the selection is the product of these two: 5 * 6 = 30.

This calculation shows all possible combinations of leads, emphasizing the independence of choices. By understanding how to multiply these independent possibilities, we can quickly calculate the probabilities involved.