LAB 7: Graphing Calculator: Calculating Permutations and Combinations

New Concept

${}_{n}P_{r}$ is read "the permutation of n things taken r at a time."

What’s next

Next, you will use a graphing calculator to find values for both permutations (${}_{n}P_{r}$) and combinations (${}_{n}C_{r}$).

The permutation function, $_{n}P_r$, calculates the number of ways to arrange 'r' objects from a set of 'n' objects, where the order of selection is important. This powerful tool, found in your calculator's MATH menu under the PRB (Probability) section, is essential for solving problems involving specific sequences, rankings, or arrangements where every position is unique.

To calculate $_{13}P_7$: Enter 13, press MATH, go to PRB, select 2: $_{n}P_r$, enter 7, and press ENTER to get 8,648,640.
To calculate $_{25}P_4$: Enter 25, press MATH, go to PRB, select 2: $_{n}P_r$, enter 4, and press ENTER to get 303,600.
How many ways to award gold, silver, and bronze to 10 racers? Calculate $_{10}P_3$ to find there are 720 ways.

Think of it like picking a president, vice president, and treasurer from a club of 18 people. The order you pick them in creates a different outcome! This calculator shortcut for $_{n}P_r$ saves you from doing massive multiplication, getting you the answer in just a few clicks. It's your secret weapon for order-based problems.

The combination function, $_{n}C_r$, calculates the number of ways to choose 'r' objects from a set of 'n' objects when the order of selection does not matter. You can easily access this function on a graphing calculator by navigating to the MATH menu and selecting the PRB (Probability) option to solve problems involving unordered groups.

To calculate $_{15}C_7$: Enter 15, press MATH, go to PRB, select 3: $_{n}C_r$, enter 7, and press ENTER to get 6,435.
To calculate $_{20}C_4$: Enter 20, press MATH, go to PRB, select 3: $_{n}C_r$, enter 4, and press ENTER to get 4,845.
How many different 4-person teams can be formed from 10 players? Calculate $_{10}C_4$ to find there are 210 possible teams.

Imagine you're picking 3 friends from a group of 14 to go to the movies. It doesn't matter if you pick Alice, then Bob, then Charlie—it's the same group going! The $_{n}C_r$ function is perfect for these 'order-doesn't-matter' scenarios, like picking a committee or a hand of cards. It does the hard work for you.

The notation $_{n}P_r$ is formally read as 'the permutation of n things taken r at a time,' emphasizing that order is critical. In contrast, $_{n}C_r$ is read as 'the combination of n things taken r at a time,' which signifies that you are simply choosing an unordered group. Understanding this language is key to interpreting problems correctly.

The expression $_{8}P_3$ is read as 'the permutation of 8 things taken 3 at a time.'
The expression $_{10}C_4$ is read as 'the combination of 10 things taken 4 at a time.'
A problem asking for 'unique arrangements' implies a permutation ($_{n}P_r$), while 'forming a committee' implies a combination ($_{n}C_r$).

Don't let the fancy words trip you up! 'Permutation' is your keyword for problems where position or order matters, like arranging letters in a word. 'Combination' is your go-to for situations where you're just forming a group, like picking toppings for a pizza. Knowing the lingo helps you choose the right calculator button every time.