Combinations with Repetition - A Shortcut

You may have had a math problem similar to this one:

Let us say there are five flavors of ice cream: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. How many variations will there be?

The solution your math teacher will give you lies in the following formula:






Where n is the 5 available flavors, and r is the 3 scoops you can have. (You should already know what a factorial (!) is. If you don't, go read up on it before you continue reading this. Don't worry, this blog won't go away...).

So the answer is 35.

But what if there were 100 ice cream choices instead of 5, and you had only 2 minutes to answer the question? It would take forever to find the factorials for the numerator and denominator.

The shortcut is to multiply only 3 numbers of the numerator, and 3 of the denominator (representing the three r choices), since the numerator and denominator cancel out much of the work automaticaly.

For example, instead of





You can cancel out half the equation,




leaving you with:





So if there were 100 ice cream choices, I would first figure out the numerator (n + r - 1) as
100 + 3 - 1 = 102
Then 102 x 101 x 100 = 1,030,200 as the numerator.
And 3 x 2 x 1 = 6 as my denominator



If you'd like to try other problems and check your answers, visit Casio's online calculator.