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 the cars can leave the intersection.

Step 1: Identify Cars as Objects Permutation

Each car can be considered a distinct object that needs to pass through the intersection. Therefore, we need to find the number of ways to arrange these objects.

Step 2: Apply Factorial Concept

Since the cars are distinct, the number of ways they can leave the intersection is the same as the number of permutations of four distinct items. This is calculated using a factorial: \(4!\).

Step 3: Calculate 4!

The factorial \(4!\) means multiplying all whole numbers from 4 down to 1. Thus, \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Conclusion

This calculation implies that there are 24 different sequences in which the cars can leave the intersection.

Key Concepts

Understanding Factorial

A factorial, often denoted with an exclamation mark "!", is a fundamental concept in combinatorics used to describe the product of an integer and all the integers below it. For example, with the number 4, the factorial of 4, or \(4!\), results in multiplying 4 by all the lower numbers until 1: \(4 \times 3 \times 2 \times 1 = 24\).
Factorials grow very quickly—while \(5!\) is already 120, and \(10!\) leaps to 3,628,800. Understanding factorials is crucial in calculating permutations, where you're interested in how to arrange objects. This mechanism simplifies the understanding of more complex arrangements, allowing easier tallying of possible configurations or sequences.

Exploring Permutations

Permutations refer to the different ways you can arrange a set of distinct objects. If you have a unique set of objects, the number of permutations tells you how many different ways you can reorder them. In the car intersection problem, because all the cars are distinct, we use permutations to find out all the possible orders they can pass through the intersection.
To determine the permutations, we use the factorial of the number of objects. For four cars, this becomes \(4!\) (as introduced in the factorial explanation), which computes the arrangement count. Fundamentally, permutations account for every minute switch in order, giving us a comprehensive understanding of all potential sequences.

Significance of Distinct Objects

In combinatorics, treating objects as distinct is important when you're looking to understand permutations and arrangements. When the objects—like the cars in the intersection problem—are distinct, each holds its position in any sequence, meaning the order matters.
Distinctness ensures that every rearrangement results in a uniquely identifiable sequence or outcome. For example, if one of our four cars was indistinguishable from another, our calculation would adjust as some permutations would be identical, and we would need to compensate for these repeats. Thus, distinct objects simplify the math, as each permutation leads to a genuinely unique configuration.