计数的基本原理是什么？

Fundamental Counting Principle

You and your friend are in the same class and you have a quiz coming up. Going in, you know that the quiz is going to be $10$ multiple choice questions with $4$ options for each. You, being the perfect student you are, have studied to prepare for the upcoming quiz, but your friend is not nearly as studious and decides to just guess on every answer on the quiz.

You have some free time now that you've gotten your studying out of the way, so you want to calculate the chance your friend has to correctly guess every question on the quiz. This is done by finding the total number of combinations of answers, and then taking the reciprocal to find the one combination that is entirely correct.

You can use the fundamental counting principle to easily calculate this. All you need to do is multiply the number of options in each level of the problem. In this case, there are $10$ problems on the quiz with $4$ answers each, so we need to multiply $4$ $10$ times. $$\begin{align}c&=4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\\ c&=4^{10}\\ c&=1048576\end{align}$$ There are over $1$ million combinations of answers on this test! Your friend has a $\frac{1}{1048576}$ or $0.000000954 \%$ chance of correctly guessing every question on the test.

Definition

The definition of the fundamental counting theory is as follows:

"If you have an event $a$ and another event $b$ then all the different outcomes for the events is $a \cdot b$."

This extends to any amount of events. For instance, if we had $4$ events, they could be named $a$, $b$, $c$, and $d$, then we could find the total number of outcomes with $a\cdot b\cdot c\cdot d$.

Examples and Practice Question

We've already used the example of a quiz, but let's use another classic example: ice cream.

Say you're looking at a menu at your favorite ice cream truck and you can choose one option from each of the following categories:

Type: hard, soft serve

Cones: regular, waffle, cup

Flavors: chocolate, vanilla, birthday cake, pistachio, strawberry

Toppings: jimmies, nuts, hot fudge, peanut butter, caramel

How many total combinations of ice cream are there? We have $4$ events: Type, cone, flavor, and toppings. Each event has a different number of options. There are $2$ types, $3$ cones, $5$ flavors, and $5$ toppings. We need to multiply these numbers together to get the total number of combinations: $$2\cdot 3\cdot 5\cdot 5=150$$ There are $150$ different combinations of ice cream options for you to pick from!

Try one for yourself:

We have $4$ categories here: medium, game type, number of players, and game length. Each of these categories have a different number of options. There are $3$ mediums, $5$ game types, $3$ player sets, and $4$ game lengths. We need to multiply all of these together to find the different combinations: $$3\cdot 5\cdot 3\cdot 4=180$$ There are $180$ different combinations of attributes.

The Fundamental Counting Principle

The Fundamental Counting Principle seems very straightforward, but it is one of the foundational theories in probability. It is a quick way to find the total number of possible outcomes when there are several choices to make.

The Fundamental Counting Principle is one way to count possible outcomes. Combinatorics with Pascal’s Triangle provides another counting method when choosing groups of objects.

Instead of listing every possible outcome, you can multiply the number of choices at each step.

Imagine that you are choosing an outfit with three shirts and two pairs of pants. Each of the three shirts can be worn with either pair of paints. To find the total number of outfits, multiply $3 \times 2$. This gives you six possible outfits.

The same idea works when there are more than two choices. Suppose a restaurant offers four main dishes, three side dishes, and two drinks. You can find the number of possible meals by multiplying $4 \times 3 \times 2$. There are $24$ different meals you could create.

When arranging several different objects, the Fundamental Counting Principle leads naturally to factorials. Factorials provide a shorter way to write a product of consecutive positive integers.

The Fundamental Counting Principle works because each choice can be paired with every choice that comes after it. It is especially helpful when listing all the possibilities would take too much time. You can use it for outfits, meals, passwords, schedules, and many other situations.

The important step is to identify how many choices are available at each stage. Then multiply those numbers together to find the total number of possible outcomes. The Fundamental Counting Principle makes counting large groups of possibilities much faster and easier.

The Fundamental Counting Principle can be used to count arrangements step by step. Permutations provide a formula for counting these arrangements more efficiently.

A tree diagram for number of outcomes shows each possible choice at every stage. The Fundamental Counting Principle provides a faster way to count the same outcomes.

(Video) Fundamental Counting Principle

by Mario's Math Tutoring

In this video by Mario's Math Tutoring, you will learn about the Fundamental Counting Principle and how to use it.