Tuesday, June 25, 2019

Season 8 - Counting Combinations

Season 8 is almost here! This season has the biggest changes yet: all 16 housemates are sexually fluid, meaning anyone else in the house could be a perfect match.

I'm going to explain how to figure out how many possible combinations of contestants there are for the matchup ceremonies this season.

(all images from MTV website, used for informational purposes only)

Two Housemates


Let's start with an easy thing to prove. If there are only two housemates, you only have one possible combination:


2 housemates = 1 combination

Four Housemates


Now let's add two new housemates, Max and Nour:

4 housemates = ? combinations

We know that Nour could be perfect matches with any one of the three other housemates. When we pair up Nour with a housemate, we know that there are two housemates left over that only has one combination. So there are only three possible combinations for four housemates:

2 housemates = 1 combination

2 housemates = 1 combination

2 housemates = 1 combination
4 housemates = 3 combinations


Six Housemates


Now let's add two more and go to six:


6 housemates = ? combinations

Let's focus on the two new additions, Kylie and Justin. If we were to assume they were a perfect match, then the four housemates left over are the same as before:


4 housemates = 3 combinations

If they weren't a perfect match, though, Kylie's partner could be anyone. Each time we partner Kylie with someone, we see that there are four housemates remaining:

4 housemates = 3 combinations

4 housemates = 3 combinations

4 housemates = 3 combinations

4 housemates = 3 combinations
So Kylie has 5 possible partners, and whenever she's partnered with a housemate, the four remaining housemates can form 3 combinations. 5 x 3...
6 housemates = 15 combinations


The Proof


Did you notice the pattern? Whenever you add two new housemates, all you have to do is pair off one of those housemates with each person in the house, and then multiply that by the number of combinations of the leftover people.

The way we might express this mathematically is in the format f(n), where f() stands for a mathematical function and n is the number of housemates. The formula is thus:

f(n) = (n - 1) x f(n - 2)

You also have to pair this with the base we started with, that two people can only form one combination of couples:

f(2) = 1

Then, if you follow this formula all the way out... you find that 16 housemates can make 2,027,025 possible matchup combinations. This process is what's called a proof by induction.

Housemates
Combinations
2
1
4
3
6
15
8
105
10
945
12
10,395
14
135,135
16
2,027,025