Probability and Statistics

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Abhinav

1:27 AM

@SiongThyeGoh math.stackexchange.com/… Can you tell me why there is (mn)! instead of $(n!)^m$like @Harambe said

Siong Thye Goh

1:46 AM

somehow i can't see the link

Abhinav

Q: Grouping items into groups

This is a chug-plug formula given in my book : 1.The number of ways in which mn different items can be divided equally int m groups, each containing n objects and the order of the groups is not important,is : $\frac {(mn)!}{(n!)^m} \frac {1}{m!} $ 2.The number of ways in which mn different item...

combinatorics

Siong Thye Goh

2:16 AM

do u mean you want to replace $(mn)!$ of the formula with $(n!)^m$?

Abhinav

yeah

Siong Thye Goh

hmmm... what is the mechanism? of constructing $(n!)^m$?

Abhinav

Since there are m groups so in any group there can be n items and we can also arrange them

Siong Thye Goh

you mean you want to replace of the formula of $\frac{(mn)!}{(n!)^m}\frac{1}{m!}$ with $\frac1{m!}$ right?

Abhinav

Mostly I am confused about how (mn)! came to existence here so it will be awesome if you could explain to me that part.

yeah

Siong Thye Goh

2:26 AM

oh, since we have $mn$ objects, we first arrange them in a straight line

after which, we consider the first $n$ items as first group, the second $n$ items as the second group and so on.

we do not care about the order within each group, so we divide $n!$ $m$ times

also, we do not care about the ordering of the groups, there are $m$ groups, so we divide by $m!$

Abhinav

3:10 AM

@SiongThyeGoh So we arrange the items first and then apply the condition for this grouping.Makes sense now

Siong Thye Goh

3:27 AM

yay.

Abhinav

one last doubt...How do we choose the groups like in this case...like we first assumed the first group to contain the first n objects etc but later changed that too

Siong Thye Goh

once we arrange them and split them into m groups, you have the grouping. the division is to avoid duplicate counting

Abhinav

So in short we get the general formulae from this case...right?

Siong Thye Goh

the general formula comes from the strategy to obtain the grouping and remove the duplicate counting

Abhinav

I get it now.Thanks for the help (^_^)

Siong Thye Goh

3:36 AM

welcome

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