Beach volleyball - how many combinations of 5 pairs can I make from 10

Beach Coach · 2017-02-07 16:00:31

I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

And if you're able to answer that one, here's a further question. We actually have 13 athletes on the roster. 10 of them can play in a match (5 doubles pairs). How many possible combinations of 5 pairs can I make from 13 athletes?

Thank you!!

bobbym · 2017-02-07 18:24:28

Hi;

I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

iamaditya · 2017-02-08 00:03:24

Hi;

Mathegocart · 2017-02-08 14:17:14

bobbym wrote:

Hi;
I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

Another GF?

bobbym · 2017-02-08 14:18:57

Nope, this one was entirely done by a very simple tool used in EM. It is called "Spot the Pattern."

Mathegocart · 2017-02-08 14:27:36

wait a moment... i see a pattern!

bobbym · 2017-02-08 14:31:23

Mathegocart · 2017-02-08 14:34:17

Last edited by Mathegocart (2017-02-08 14:35:00)

bobbym · 2017-02-08 14:39:20

Hi;

Beach Coach · 2017-02-09 09:18:42

I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

Here's more detail:

There are 10 athletes.
We make 5 pairs to play in matches.
Any of the 10 athletes can be paired with any of the other 9 to make a pair.
There are always 5 pair.

So far, the answers posted are 45, 250 and 945.

Having made these pairs for 10 seasons, I've realized there are a lot of possible pairings. I was thinking it would be 5,000 to 6,500 or around 25,000. I'm a Harvard grad but not a math whiz, so I appreciate your input greatly.

If you'll share your formula so all who are watching can help, that would be great.

Thank you to all of you who have helped so far.

bobbym · 2017-02-09 12:53:01

Hi coach;

I am sure somebody can come up with a formula that will spit out 25000 or any other number you want but until someone shows me a program that counts more than 945 ways, I am going to stick with that.

phrontister · 2017-02-12 00:47:55

Hi;

I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

I got 945 (as Bobby did), also by playing "Spot the Pattern".

OEIS gives that answer too, and the formula (2*n)!/(n!*2^n). In our case n is the number of pairs of athletes.

The answer is the product of the first n odd numbers: 1*3*5*7*9 = 945.

I did the first four elements of the pattern by hand, but stopped there to play STP as the fifth was going to mean I'd be writing into the middle of next week!

bobbym · 2017-02-12 01:48:34

Hi;

Yes! STP! And for backup...OEIS! Wunderbar!

phrontister · 2017-02-12 01:54:01

How else!

Here are images of my work for the STP:

Last edited by phrontister (2017-02-23 00:01:08)

bobbym · 2017-02-12 02:00:22

How else!

Did you try a bit of M?

phrontister · 2017-02-12 02:14:39

Yes I did, but didn't get there.

I've been looking at Subsets[Range], and also DeleteDuplicates[Sort/@Permutations[Range[]]]. Still looking at it...

Last edited by phrontister (2017-02-12 02:40:32)

bobbym · 2017-02-12 02:27:02

It is a bit tricky.

phrontister · 2017-02-12 02:28:20

I'll say!

Still looking...

Any hints?

bobbym · 2017-02-12 02:32:37

Yes, when you get the answer save the notebook! I deleted mine and now can not duplicate my work.

phrontister · 2017-02-12 02:38:05

Not quite the kind of hint I was looking for, but good advice!

Maybe a data recovery program could find the deleted file...if it hasn't become too scrambled over time.

bobbym · 2017-02-12 02:52:28

Never had much luck with those.

Try this:

n = 1;
s = Permutations[Range[2 n]];
teams = Partition[#, 2] & /@ s;
DeleteDuplicates[Map[Sort, teams, 2]] // Length

Adjust n for higher values. Suffers from combinatorial explosion but is short.

phrontister · 2017-02-12 03:45:19

Yes, that's it! I've saved it, so if you ever need a copy...!

Not being able to make pairs had been holding me up, but I hadn't thought of using Partition. Never used it before.

I haven't tried problem #2 (the one about 13 athletes) in the first post yet, but I'm off to bedzzzzzz....

Catch you later.

bobbym · 2017-02-12 03:46:19

Hi;

Okay, see you later.

phrontister · 2017-02-13 03:43:22

Beach Coach wrote:

We actually have 13 athletes on the roster. 10 of them can play in a match (5 doubles pairs). How many possible combinations of 5 pairs can I make from 13 athletes?

270270, I think.

I played STP by hand in Excel for levels 1 & 2, but the size of the problem was escalating enormously and I had to change tack. So then I adapted Bobby's M code from post #21 to fit my levels 1 & 2 results and got levels 3 & 4 (but then ran out of computer memory, preventing further progress that way).

At that stage I had the sequence {10,105,1260,17325}, covering the first 4 levels.

At OEIS I found the last (5th) level and the following formula for the sequence:

(2n+3)!/(3!*n!*2^n)

For our problem, n is the number of doubles pairs (5), from which (2*5+3)!/(3!*5!*2^5) = 270270.

Here are images of my work for the STP:

Last edited by phrontister (2017-02-23 00:02:44)

Beach Coach · 2017-02-13 18:24:07

First of all, THANK YOU for all this great input and thinking!

So, is this the consensus:

I coach beach volleyball. We field 5 pairs of athletes. How many possible combinations of 5 pairs I can make from 10 athletes?

ANSWER: (2*n)!/(n!*2^n). In our case n is the number of pairs of athletes. The answer is the product of the first n odd numbers: 1*3*5*7*9 = 945.

Here's a further question. We actually have 13 athletes on the roster. 10 of them can play in a match (5 doubles pairs). How many possible combinations of 5 pairs can I make from 13 athletes?

ANSWER: (2n+3)!/(3!*n!*2^n) - For our problem, n is the number of doubles pairs (5), from which (2*5+3)!/(3!*5!*2^5) = 270270.

Math Is Fun Forum

#1 2017-02-07 16:00:31

Beach volleyball - how many combinations of 5 pairs can I make from 10

#2 2017-02-07 18:24:28

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#3 2017-02-08 00:03:24

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#4 2017-02-08 14:17:14

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#5 2017-02-08 14:18:57

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#6 2017-02-08 14:27:36

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#7 2017-02-08 14:31:23

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#8 2017-02-08 14:34:17

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#9 2017-02-08 14:39:20

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#10 2017-02-09 09:18:42

Re: Beach volleyball - how many combinations of 5 pairs can I make from 10

#11 2017-02-09 12:53:01

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#12 2017-02-12 00:47:55

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#13 2017-02-12 01:48:34

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#14 2017-02-12 01:54:01

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#15 2017-02-12 02:00:22

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#16 2017-02-12 02:14:39

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#17 2017-02-12 02:27:02

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#18 2017-02-12 02:28:20

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#19 2017-02-12 02:32:37

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#20 2017-02-12 02:38:05

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#21 2017-02-12 02:52:28

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#22 2017-02-12 03:45:19

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#23 2017-02-12 03:46:19

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#24 2017-02-13 03:43:22

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#25 2017-02-13 18:24:07

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