36 University

New ACT Math Test Content Part 1: The Factorial

Combinatorics has been added to the ACT math test—and 36U is here to get you ready!

Don’t worry. Combinatorics may sound intimidating, but it’s really about fancy, advanced counting techniques that can be a lot of fun and save lots of time. Let’s get right to it.

Factorials

First, let’s work on a possibly new concept with some new notation—the factorial.

6 factorial, written 6!, means 6 • 5 • 4 • 3 • 2 • 1.

And, again:

5! = 5 • 4 • 3 • 2 • 1

To simplify, 5! = 5 • 4 • 3 • 2 • 1 = 120.

Let’s put the factorial to work…

Example 1:  In how many different ways can you arrange the letters of the word OVERCAST?

____  ____  ____  ____  ____  ____  ____  ____

You have 8 options for the first letter, 7 options for the 2^nd, 6 options for the 3^rd, and so on.

Mathematically, that looks like this:

_8 options  •_7 options  •_6 options  •_5 options  •_4 options  •_3 options  •_2 options  •_1 option 

or

8! –>  40,320

There are 40,320 different ways the letters of OVERCAST can be arranged.

Easy so far. Let’s move on…

Example 2: In how many different ways can you arrange the letters of the word CANYON?

This is a slightly trickier item because CANYON has 2 N’s!  Fortunately, if you understood our first example, this isn’t much more difficult.

Step 1: There are six different spots to place the letters in the word CANYON:

_6 options  •_5 options  •_4 options  •_3 options  •_2 options  •_1 option     =  6!

There are 6! or 720 different ways of arranging C-A-N-Y-O-N, but…You have to account for repeated options because there are 2 Ns. Here’s how:

Take your total number of possible arrangements (6!) and divide by 2! to account for N appearing twice.

6!/2! –>  (6 • 5 • 4 • 3 • 2 • 1)/(2 • 1)

–>  360

There are 360 different ways the letters of the word CANYON can be arranged!

That’s your introduction to new counting techniques (combinatorics) that are being tested on the ACT.

Take time to brush up on your combination and permutations, too. For more instruction and practice on these topics, check out our online program.

-Dr. Kendal Shipley, 36U

10/13/17