devoted to the study of temari and mathematics
Using a basic mathematical formula for the number of combinations in a set of objects you can calculate how many ways there are to combine the five Platonic solids into smaller groups. Once you know how many there are it is a straight forward task to list them all and know that you have not left any out.
Before we tackle the question of the Platonics, let's look at a similar problem. Consider that you have a set of 5 colors: blue, red, yellow, green, purple. You would like to know how many different ways you can put three colors together. Notice that blue, red, green is really the same as red, green, blue. In this case the order does not matter. Mathematicians call that a combination and use a formula to calculate the number of possibilities. It looks like this:
nCk = n! ÷ [k! (n-k)!]
The n represents the number of items in the original set and the k represents the number of items you want to combine. The C is really the name of the formula...Combinations. You would say the number of combinations of n items taken k at a time.
There is a special operator here that you may not be familiar with. It is represented with an exclamation point and is called a factorial. When you have something like 8! it means to start at 8 and multiply all of the numbers together as you count down to 1. So 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320. Zero is a special case for factorials and is defined to be 1. That is, 0! = 1 (If you are curious about why that is so, visit the MathWorld definition page.)
Now to solve the problem all we need to do is put the correct numbers into the equation and do a lot of arithmetic to calculate the answer. In the case of our color example we want 5 things taken 3 at a time or:
5C3 = 5! ÷ [3! (5-3)!]
You can multiply all of the factorials or you can use algebra skills to simplify it before you multiply but in any case the final answer is 10. There are 10 ways to take three colors out of the original set of five.
We want to calculate all of the ways to combine the Platonic Solids. We know that we have a set of 5 objects. We'll need to calculate each of the different size groups separately. The case of 5 things taken 5 at a time is really trivial (there's only 1 way to do it), but I included the formula for completeness sake.
| Group size | English description | Formula | Answer |
|---|---|---|---|
| 2 | combinations of 5 things taken 2 at a time | 5C2 = 5! ÷ [2! (5-2)!] | 10 |
| 3 | combinations of 5 things taken 3 at a time | 5C3 = 5! ÷ [3! (5-3)!] | 10 |
| 4 | combinations of 5 things taken 4 at a time | 5C4 = 5! ÷ [4! (5-4)!] | 5 |
| 5 | combinations of 5 things taken 5 at a time | 5C5 = 5! ÷ [5! (5-5)!] | 1 |
So, the total number of ways to combine the Platonics is 10+10+5+1=26. Note that this method does not allow for repeating any element. If we want to create groups with the possibility of repeating elements that is a whole new ball game and a bit more complicated.
Now that we know how many there are it is fairly quick to list the possibilities and be sure that we have not missed any.