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 |
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
|
mathematically it's starts with less combinations than other seasons so should be easier i think
ReplyDeletePotentially easier - depends if they change the episodes or the rules. Blackout odds increase substantially
DeleteThe Function is also
ReplyDeletef(n) = (n!)/{[(n/2)!][2^(n/2)]}
Where n is the number of contestant and n needs to be an even positive integer.
So if we had it 22 contestants with boy girl matches it was 11! = 39,916,800 possibilities
but if we go with this new formula with 11 contestants we would have 13,749,310,575 possibilities
Basically it comes from the idea of choosing pairs. So you 'choose' 2 from n people. Which is "n choose 2" = n!/[2!*(n-2)!]
Then the next pair is "(n-2) choose 2" where n-2 is remaining people not chosen the previously and so on... in the end you will have n/2 pairs (i.e. 16 people 8 couples) and then those n/2 couples can be arranged n/2 ways i.e. (n/2)!
So you take
(the ways pairs can be selected i.e. "n Choose 2" * "(n-1) Choose 2" * ... * "2 choose 2")
and divide by (the ways they can be arranged = (n/2)! )
Then you end up using the formula, with the same result of 2,027,025 with 16 people, 8 matches. Hopefully that is insightful for some people. Anyways...I'm looking forward to this season.
Very cool! Thanks for sharing!
DeleteThis comment has been removed by the author.
ReplyDeleteCorrection 1: 22 contestants (11 Pairs) we would have 13,749,310,575 possibilities
ReplyDelete2: "n Choose 2" * "(n-2) Choose 2" * ... * "2 choose 2")
Thank you thank you thank you for continuing this blog! I am so glad you're still doing it, I love coming here every week and looking at the percentages.
ReplyDeleteExcellent blog!
ReplyDeletethanks for coming back! gonna be an interesting season. at least they have Danny to figure it out.
ReplyDeleteHey Alex - caught your name in VICE. Hoping to reach you for a story for ELLE.com. Could you shoot me an email? rose.minutaglio@gmail.com
ReplyDeleteHi! I'm just wondering how you calculate percentages after match-up ceremonies, or how that goes into the calculation--just very interested in the math behind this :)
ReplyDeleteHello,
ReplyDeleteJonathan - Basit
Kai - Danny
Brandon - Aasha
Jasmine - Nour
Jenna - Paige
Kari - Max
Kylie - Amber
Justin - Remy
Correct me if I'm wrong.
Could someone explain how Jasmine and Nour are a match? I understand it's because Brandon and Aasha are a match, but I don't understand the correlation between the two.
ReplyDeleteWith all the information given, you can deduce all the perfect matches.
ReplyDeleteConfirmed Matches: Brandon and Aasha, Jonathan and Basit
Episode 4: 1 beam: Max and Justin
All other couples on week 4 are either confirmed no match, confirmed via blackout, and were not paired up on week 1
Episode 3: 2 beams: Amber and Paige, Jasmine and Nour
All other matches were incorrect from confirmed no matches
Danny and Kylie: Kylie is the only remaining possible option for Danny
Remy and Kai: Kai is the only remaining possible option for Remy
Last Couple: Jenna and Kari
thank you!!
ReplyDeletei'm learning about probability in my phd and i love this show - this blog is perfect