Lesson 111: Solving Problems Involving Permutations

New Concept

The number of permutations of $n$ objects taken $r$ at a time is given by the formula:

What’s next

Next, you’ll apply the permutation formula and related counting principles to solve problems involving arrangements, from class schedules to competition outcomes.

Property

If an independent event $M$ can occur in $m$ ways and another independent event $N$ can occur in $n$ ways, then the number of ways that both events can occur is $m \cdot n$.

Explanation

Imagine you're building a video game character. If you have 6 hairstyles and 4 outfits, how many unique looks can you create? Instead of listing them all out, just multiply! This principle is your ultimate shortcut for finding total combinations when you have multiple independent choices to make, saving you from drawing a massive tree diagram.

Examples

• A cafe offers 5 sandwich types and 3 different drinks. The total number of meal deals is $5 \times 3 = 15$.
• You have 2 choices for a departure flight and 4 choices for a return flight. You can schedule your trip in $2 \times 4 = 8$ different ways.
• For a simple password, you need one letter (26 options) and one digit (10 options). The total number of passwords is $26 \times 10 = 260$.

Let's see how quickly we can count all the options without listing a single one. This example applies the Fundamental Counting Principle.

Example Problem
A coffee shop offers a choice of 5 types of coffee beans and 3 types of milk. Find the number of ways a one-bean, one-milk coffee can be ordered.

Step-by-Step

1. Determine the number of ways each independent event can occur. There are 5 choices for coffee beans and 3 choices for milk.
2. According to the Fundamental Counting Principle, we find the product of the number of ways for each event.

This demonstrates there are 15 possible coffee combinations.

Property

The factorial $n!$ is defined for any natural number $n$ as $n! = n(n-1)...(2)(1)$. Zero factorial is defined to be 1. $0! = 1$.

Explanation

Factorials are all about arranging things in a line. If you have 'n' items, you have 'n' choices for the first spot, then 'n-1' for the second, and so on down to one. It's like a countdown multiplication party! It’s the perfect tool for calculating the total number of ways to order an entire group.

Examples

• To find the value of five factorial, you multiply all whole numbers from 5 down to 1: $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$.
• To simplify a factorial fraction, cancel out the common parts: $\frac{8!}{5!} = \frac{8 \cdot 7 \cdot 6 \cdot 5!}{5!} = 8 \cdot 7 \cdot 6 = 336$.
• The number of ways 4 friends can stand in a line for a photo is $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$ ways.

Property

The number of permutations of $n$ objects taken $r$ at a time is given by the formula $_nP_r = \frac{n!}{(n-r)!}$.

Explanation

Permutations are for when order is king, like winning 1st, 2nd, and 3rd place in a race. You have a big group of 'n' things, but you're only picking and arranging 'r' of them. This formula is your secret weapon to calculate all possible ordered arrangements without having to list them all out one by one.

Examples

• In a race with 8 runners, the number of ways to award gold, silver, and bronze medals is $_8P_3 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 336$.
• From a team of 10 players, the number of ways to pick a captain and a vice-captain is $_{10}P_2 = \frac{10!}{(10-2)!} = \frac{10!}{8!} = 90$.
• The number of ways 5 different books can be arranged on a shelf is $_5P_5 = \frac{5!}{(5-5)!} = \frac{5!}{0!} = 120$.

When the order of finishers matters, the number of possibilities changes. Let's calculate it using the core idea of permutations.

Example Problem
In a race with 10 runners, how many different ways can the first, second, and third-place medals be awarded?

Step-by-Step

1. This is a permutation because the order in which the runners finish is important. We need to find the number of permutations of 10 runners taken 3 at a time.
2. Write the permutation formula: $_nP_r = \frac{n!}{(n-r)!}$.
3. Substitute $n=10$ and $r=3$ into the formula to get the simplified factorial expression.

4. Now, write out the factors for each factorial. Then, cancel the common terms to simplify.

5. Multiply the remaining numbers to find the final count.

There are 720 different ways to award the top three medals.