Fundamental Counting Principle

The fundamental counting principle is a rule used to count the total number of possible outcomes in a situation. It states that if there are $n$ ways of doing something, and $m$ ways of doing another thing after that, then there are $n\times m$ ways to perform both of these actions. In other words, when choosing an option for $n$ and an option for $m$, there are $n\times m$ different ways to do both actions.

Lily is trying to decide what to wear. She has shirts in the following colors: red, purple, and blue, and she has pants in the following colors: black and white. How many different outfits can Lily choose from (assuming she selects one shirt and one pair of pants)?

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We know from the definition of the rule of product that if there are $n$ options for doing one thing (like choosing a shirt), and $m$ options for doing another thing (like choosing a pair of pants), then there are $n \times m$ total combinations we can choose from. In this case, there are $3$ options for choosing a shirt, and there are $2$ options for choosing pants. Thus, there are $3 \times 2 = 6$ total options.

Here is a table where each row represents a possible outfit.

Shirt	Pants
Red	Black
Blue	Black
Purple	Black
Red	White
Blue	White
Purple	White

As expected, there are $6$ possible combinations. $_\square$

In the example above, there were two things to choose: a shirt and a pair of pants. However, the rule of product can extend to however many things to choose from. For example, if there are $n$ choices for a shirt, $m$ choices for a pair of pants, $x$ choices for a pair of shoes, and $y$ choices for a hat, the rule of product states that there are $n \times m \times x \times y$ total possible combinations.

There are $8$ daily newspapers and $5$ weekly magazines published in Chicago. If Colin wants to subscribe to exactly one daily newspaper and one weekly magazine, how many different choices does he have?

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Colin has $8\times5=40$ choices. $_\square$

Calvin wants to go to Milwaukee. He can choose from $3$ bus services or $2$ train services to head from home to downtown Chicago. From there, he can choose from 2 bus services or 3 train services to head to Milwaukee. How many ways are there for him to get to Milwaukee?

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Since Calvin can either take a bus or a train downtown , he has $3+2 =5$ ways to head downtown (Rule of sum). After which, he can either take a bus or a train to Milwaukee, and hence he has another $2+3=5$ ways to head to Milwaukee (Rule of sum). Thus in total, he has $5 \times 5 = 25$ ways to head from home to Milwaukee (Rule of product). $_\square$

Six friends Andy, Bandy, Candy, Dandy, Endy, and Fandy want to sit in a row at the cinema. If there are only six seats available, how many ways can we seat these friends?

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For the first seat, we have a choice of any of the 6 friends. After seating the first person, for the second seat, we have a choice of any of the remaining 5 friends. After seating the second person, for the third seat, we have a choice of any of the remaining 4 friends. After seating the third person, for the fourth seat, we have a choice of any of the remaining 3 friends. After seating the fourth person, for the fifth seat, we have a choice of any of the remaining 2 friends. After seating the fifth person, for the sixth seat, we have a choice of only 1 of the remaining friends. Hence, by the rule of product, there are $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ ways to seat these 6 people. More generally, this problem is known as a permutation. There are $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$ ways to seat $n$ people in a row. $_\square$

How many positive divisors does $2000 = 2^4 5^3$ have?

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Any positive divisor of 2000 must have the form $2^a 5^b$, where $a$ and $b$ are integers satisfying $0 \leq a \leq 4, 0 \leq b \leq 3$. There are 5 possibilities for $a$ and 4 possibilities for $b$, hence there are $5 \times 4 = 20$ (rule of product) positive divisors of 2000 in all. $_\square$

• Problem solving

• Rule of Sum
• Permutations