Problem 32 - Algebra 2

Question

The manager of a four-screen movie theater is deciding which of 12 available movies to show. The screens are in rooms with different seating capacities. How many ways can she show four different movies on the screens?

Step-by-Step Solution

Answer

There are 11,880 ways to show four different movies on the screens.

Step 1: Understanding the Problem

The problem involves choosing 4 different movies from a total of 12 available movies to show on 4 screens, each with a unique seating capacity. This means the order in which movies are chosen matters because each screen can be considered unique in terms of capacity.

Step 2: Choosing 4 Movies

First, we need to choose 4 movies from 12 available. The number of combinations of 4 movies from 12 is given by the combination formula: \( \binom{12}{4} = \frac{12!}{4!(12-4)!} \).

Step 3: Calculating Combinations

Calculate \( \binom{12}{4} \) which simplifies to \( \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \).

Step 4: Assigning to Screens

After choosing 4 movies, we assign them to 4 screens. Since each screen is different, each arrangement is unique. There are 4! = 24 ways to arrange these 4 movies on 4 screens.

Step 5: Calculating Total Arrangements

Multiply the number of ways to choose movies by the number of ways to assign them: \( 495 \times 24 = 11880 \). Thus, there are 11,880 ways to show 4 different movies on the screens.

Key Concepts

Permutations

Permutations are all about arranging items. When you consider permutations, the order matters. Imagine you have a set of objects, and you want to know in how many different orders you can arrange them. This is exactly what permutations answer.
For example, if you have 4 different movies and 4 unique screens, as in the theater manager problem, each movie can be placed on any of the 4 screens. The uniqueness of the screens means every arrangement counts as different. This is why, after choosing which 4 movies are to be shown, we calculate the ways to arrange them using permutations.
To calculate the number of permutations for a set number of items, you use the factorial function—more on that later—to find all possible arrangements. Specifically, for arranging 4 movies, you calculate the permutations as 4!, which equals 24 different ways.

Combinations

Combinations are used when the order of selection doesn't matter. They help you figure out how many ways you can choose a subset from a larger set.
In our movie theater problem, the first task was to choose 4 movies out of 12 without considering the order initially. This scenario makes use of combinations because selecting which movies to show is independent of which screen they'll go to at first.
To calculate combinations, we use the formula: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \( n \) is the total number of choices, and \( r \) is the number of selections we want to make. In our example, \( n \) is 12 (the total movies), and \( r \) is 4 (movies chosen). Solving this gives us 495 ways to select which movies will be shown.

Factorial

Factorial, represented by an exclamation mark (!), is a concept that helps in calculating permutations and combinations. It is the product of all positive integers up to a certain number. For instance, 4! (read as "four factorial") means 4 × 3 × 2 × 1, which equals 24.
Factorials grow quite large quickly, so they're really useful in combinatorial calculations like the ones needed for permutations and combinations.
They help simplify expressions, especially in formulas where large products define possible selections or arrangements. For instance, in combinations, factorials help simplify the formula \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]by canceling out terms. Hence, understanding how factorial works simplifies tackling problems in combinatorics effortlessly.