Michigan League of Academic Games

Fun with Factorials! (All Adv. Equations)

One of the most popular variations in all levels of Equations is factorial. Students enjoy it because it is easy to understand, but there are a lot of different things you can do with it.

Remember that factorial is represented by a ! that may be placed anywhere in the solution or goal (elementary and middle only). Factorial is defined as the product of all the numbers between 1 and the number being acted upon. That is 5! = 5 x 4 x 3 x 2 x 1 = 120, and 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040. Note that in the junior and senior division, factorials can not be used in the Goal, just the Solution.

Here are a few interesting goal - solution combinations that you can use with factorial:

1) 17 / 16 = 17

Explanation: Goal is interpreted as 17! divided by 16!. 17! = 17x16x15x14x13... When divided by 16x15x14x13..., all the numbers cancel out and the only the 17 is left. This goal can be used with any two consecutive numbers. The higher the numbers, the more effective the goal. This strategy could not be used in the junior or senior divisions.

2) 24 x 23 = 24!

Explanation: Goal is interpreted as 24 x 23!, which equals 24x23x22x21x20..., or 24! Like the previous goal, it can be used with any two consecutive numbers. With both of these soultions, be careful not to use too many factorials in the solution. This strategy could not be used in the junior or senior divisions.

3) 54 x 56 = 9! / 5!

Explanation: Goal equals (9 x 6) x (7 x 8), or 9 x 8 x 7 x 6, which equals 9! (9x8x7x6x5x4x3x2) divided by 5!(5x4x3x2). There are a number of different number combinations that this factor rearrangement works with.

Factorial also works very well with a number of different variations, including LCM, GCF, # of factors, Powers of the Base, Multiple of K. Experiment a little in your practices, and you will probably find a few new tricks of your own.