How many ways could a class of 18 students divide into group

[wind]

How many ways could a class of 18 students divide into groups of 3 students each?

Have 18 students take 3 out, then do that 6 times?
6(18P3)

is that right?


Thanks

 

[pka]

Try \(\displaystyle \L\frac{{18!}}{{\left( {6!} \right)\left( {3!} \right)^6 }}\)

Can you explain why that works?

EDIT: I first read it as three groups not three in a group.

 

[soroban]

Re: How many ways could a class of 18 students divide into g

Hello, wind!

How many ways could a class of 18 students divide into groups of 3 students each?

I'm sure that pka meant: \(\displaystyle \L\:\frac{18!}{6!(3!)^6}\)


Let's baby-step through this problem . . .

First, we'll number the groups, 1 to 6.

For group #1, there are: \(\displaystyle \,{18 \choose 3}\) ways.
For group #2, there are: \(\displaystyle \,{15 \choose 3}\) ways.
For group #3, there are: \(\displaystyle \,{12 \choose 3}\) ways.
For group #4, there are: \(\displaystyle \,{9 \choose 3}\) ways.
For group #5, there are: \(\displaystyle \,{6 \choose 3}\) ways.
For group #6, there are: \(\displaystyle \,{3 \choose 3}\) ways.

Hence, there are: \(\displaystyle \L\:\frac{18!}{3!\sout{15!}}\cdot\frac{\sout{15!}}{3!\sout{12!}}\cdot\frac{\sout{12!}}{3!\sout{9!}}\cdot\frac{\sout{9!}}{3!\sout{6!}}\cdot\frac{\sout{6!}}{3!\sout{3!}}\cdot\frac{\sout{3!}}{3!\sout{0!}} \:=\:\frac{18!}{(3!)^6}\) ways.

But the groups are not numbered; their order is not considered.

Since our answer includes the \(\displaystyle 6!\) different orders of the six groups,
. . we must divide by \(\displaystyle 6!\)

Answer: \(\displaystyle \L\:\frac{18!}{6!(3!)^6}\)