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

15 comments:

  1. mathematically it's starts with less combinations than other seasons so should be easier i think

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    Replies
    1. Potentially easier - depends if they change the episodes or the rules. Blackout odds increase substantially

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  2. The Function is also
    f(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.

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  3. This comment has been removed by the author.

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  4. Correction 1: 22 contestants (11 Pairs) we would have 13,749,310,575 possibilities
    2: "n Choose 2" * "(n-2) Choose 2" * ... * "2 choose 2")

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  5. 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.

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  6. thanks for coming back! gonna be an interesting season. at least they have Danny to figure it out.

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  7. Hey 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

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  8. Hi! 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 :)

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  9. Hello,

    Jonathan - Basit
    Kai - Danny
    Brandon - Aasha
    Jasmine - Nour
    Jenna - Paige
    Kari - Max
    Kylie - Amber
    Justin - Remy

    Correct me if I'm wrong.

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  10. 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.

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  11. With all the information given, you can deduce all the perfect matches.
    Confirmed 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

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  12. thank you!!
    i'm learning about probability in my phd and i love this show - this blog is perfect

    ReplyDelete