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How many different groups of $4$ can I create using $24$ students?

I want to break my class of $24$ students into groups of $4$. I would like to create different groups each day until each student has worked with every other student in the class at least one time. I have tried visually charting all the students and organizing them into different groups of $4$. I am looking for an easier way to do this.

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    $\begingroup$ en.wikipedia.org/wiki/Combination $\endgroup$ – dimebucker Aug 20 '15 at 14:40
  • $\begingroup$ Are you aware of asking two different questions? $\endgroup$ – principal-ideal-domain Aug 20 '15 at 14:44
  • $\begingroup$ There is a subtle point when you want to count the number of ways to divide $24$ people into $6$ groups of $4$. Are the groups labelled (Team Red, Team Blue, and so on), or are they unlabelled? The answers are related, but quite different. $\endgroup$ – André Nicolas Aug 20 '15 at 15:03
  • $\begingroup$ "I would like to create different groups each day until they have worked with every student in the class one time." This honestly sounds like a scheduling problem, and this could easily fall under the purview of graph theory. Here is the Round Robin Algoirthm. Perhaps that will help you a bit. I'm not sure if it is exactly what you're looking for, but it may give some ideas. $\endgroup$ – Paddling Ghost Aug 20 '15 at 15:07
  • $\begingroup$ I asked a question related to this one which may address your implicit question here. If you happen to really want to make such a schedule, I would suggest following this question. $\endgroup$ – Paddling Ghost Aug 20 '15 at 17:27
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As principal-ideal-domain notes, it seems like there are two different questions that you are asking.

The first question is: How many different ways can I split my 24 students into six groups of four?

The second question is: How many different groups of four can I create out of my 24 students?

I'll start with the second question, where the link provided by dimebucker91 should be useful. Generally, if you have n students, and want to create a group of k, the general formula is:

$$ \frac{n!}{k!(n-k)!} $$

So here, you have 24 students and want to create a group of four, so you are looking for the solution to:

$$ \frac{24!}{4!(20)!} $$

The answer to the first question is an extension to the answer of the second question. If you have 24 students, and pick any given four of them to create the first group, you have the above-mentioned result. Then you do the same thing for the next four students, and pick another four from your remaining 16 students, and so on, so your result looks something like:

$$ \frac{24!}{4!(20)!} * \frac{20!}{4!(16)!} * \frac{16!}{4!(12)!} ... $$

We could multiply these out, but we might also note that many of these terms cancel, so the final result is actually just:

$$ \frac{24!}{(4!)^6} $$

which is about 3.246*10^15 combinations.

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The number of different groups of four you can get from 24 students is given by $\binom{24}4=\frac{24!}{(24-4)!\cdot 4!}=10,626$

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This is known as the Social Golfer Problem. Best you can do without repetition is 7 days.
day 1: ABKU IJSE QRCM DGFX HLNO PTVW
day 2: ACLV IKTF QSDN EHGR BMOP JUWX
day 3: ADMW ILUG QTEO FBHS CJNP KRVX
day 4: AENX IMVH QUFP GCBT DJKO LRSW
day 5: AFOR INWB QVGJ HDCU EKLP MSTX
day 6: AGPS IOXC QWHK BEDV FJLM NRTU
day 7: AHJT IPRD QXBL CFEW GKMN OSUV

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