Question 1200705: In a classroom of 24 students, the teacher forms groups of 6 students each. How many different ways can these groups be formed?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
The number of different ways to select 6 students from a group of 24 students is,
BY THE DEFINITION, the number of combinations of 24 students taken 6 at a time
 =  =  =  = 134596. ANSWER
Solved.
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On Combinations, see introductory lessons
- Introduction to Combinations
- PROOF of the formula on the number of Combinations
- Problems on Combinations
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.
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If the answer is C(24,6)*C(18,6)*C(12,6)*C(6,6), then the problem's formulation in the post is incorrect.
For this answer to be correct, the correct formulation should be as follows
In how many ways 24 students of the class can be distributed in 4 disjoint groups of 6 students in each group.
It is just a bad tradition at this forum posting wrong problems.
I always say in response, that the Math education of the problems' composers at this forum
is from reading puzzles, not from learning Math.
It looks like they are not familiar with high standards of writing Math problems, at all.
Consider to hire 1 - 3 - 5 - 7 - 9 specialists, who will edit your mathematical writing on the daily basis.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Interpret the problem differently than the other tutor; it seems to me all 24 students must be formed into groups of 6.
To form the first group, the teacher can choose any 6 of the 24 students; the number of ways to do that is "24 choose 6", C(24,6).
Then there are 18 students left with which to form the second group of 6 -- in C(18,6) ways.
And then C(12,6) ways to form the third group, and finally C(6,6) ways to form the last. Of course C(6,6) is 1, because with 6 students left there is only one way to form a group of 6.
ANSWER: C(24,6)*C(18,6)*C(12,6)*C(6,6) = ...(large number....)
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