Problem 12 - Calculus for Biology and Medicine

Question

Four cars arrive simultaneously at an intersection. Only one car can go through at a time. In how many different ways can they leave the intersection?

Step-by-Step Solution

Answer

There are 24 different ways for the cars to leave the intersection.

Step 1: Understand the Problem

Four cars need to pass through an intersection, with only one car able to go through at a time. Our task is to determine in how many different sequences the cars can pass through the intersection.

Step 2: Identify the Variables

Let's denote the cars as A, B, C, and D. We need to find all possible orders in which A, B, C, and D can go through the intersection.

Step 3: Determine the Total Number of Permutations

The number of ways to arrange 4 cars (or any 4 distinct items) is given by the factorial of 4, which is calculated as \[ n! = 4! = 4 \times 3 \times 2 \times 1 \].

Step 4: Calculate the Factorial

Compute the factorial of 4: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \].

Step 5: Conclusion

The number of different ways the four cars can leave the intersection is 24.

Key Concepts

Permutations

In combinatorics, permutations refer to the different ways you can arrange a set of objects. When considering permutations, the sequence matters greatly. If you have a collection of distinct items, their order will significantly influence the number of permutations you can achieve.
In the original exercise, the context is how four cars can proceed through an intersection. Each permutation is a unique sequence in which these cars pass, meaning the order of cars is essential.

• For instance, if you have cars labeled as A, B, C, and D, then the sequence such as ABCD is different from BACD.
• With four distinct cars, we want to calculate all possible ways we can arrange these vehicles, taking their sequence into account.

Thus, permutations of these cars determine all potential arrangements as they navigate the intersection. Recognizing permutations helps to understand that the problem isn't merely about moving cars; it's focused on the number of distinct sequences possible for movement.

Factorial

A factorial, often denoted by an exclamation point (!), is a mathematical operation used to determine the number of ways to arrange n distinct objects. It's a fundamental concept in permutations.
When we talk about a factorial, we're referring to a series of multiplications that start from the number n down to 1. Let's look at the example of 4 cars:

• To find the total number of permutations of these 4 cars, we compute the factorial of 4, represented as \( 4! \).
• This involves multiplying all positive integers up to 4: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

The result tells us there are 24 different arrangements, or sequences, in which the cars can leave the intersection one after the other. Factorials simplify solving permutation problems by giving a straightforward method to compute large numbers of arrangements.

Problem-Solving

Problem-solving involves breaking down a problem into manageable steps. With permutation problems, this means understanding the structure of the problem, defining variables, and using mathematical formulas effectively.
For our car sequencing example, here’s how problem-solving aids in finding the solution:

• First, clearly understand the constraints. Only one car can go through the intersection at a time, meaning we need the sequence of cars.
• Next, identify what you know and what you need to find. Here, knowing the number of cars (4) helps you decide on the permutation using factorials.
• Calculate the solution using the principles of permutations represented by factorial functions, ensuring you accurately count each possible sequence.

By following a structured approach, you can tackle permutation challenges effectively and reduce the complexity of seemingly intricate arrangements.