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