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Possible Combinations when Distributing Items Unevenly

Date: 05/24/2004 at 15:40:54
From: Jessica
Subject: Combinations of 20 objects for 5 children

There are 20 distinct toys to be handed out to 5 children.  How many 
possible ways are there to hand them out if two children each get 7 
toys and three children get 2 toys each?

I know that there are 10 combinations of how to arrange what child 
gets how many presents (5!/2!3!), but it's the rest I'm stuck on.  I
tried [C(20 7)+ C(20 7) + C(20 2) + C(20 2) + C(20 2)] * 10, but the
number is far too high.  Can you help?

Date: 05/24/2004 at 16:23:37
From: Doctor Vogler
Subject: Re: Combinations of 20 objects for 5 children

Hi Jessica,

What you need are the multinomial coefficients.  These are a 
generalization (or an extension) of the binomial coefficients (also
known as combinations, or n-choose-r).

If you have n objects, and you want to put them all into k boxes, so
that box i (for i=1,2,...k) gets a_i objects (which means that the sum
of all of the a_i's has to be n), then you can do this in


ways.  In your case, you have n=20 objects (toys), and k=5 boxes 
(children), who get a_1 = 7, a_2 = 7, a_3 = 2, a_4 = 2, a_5 = 2
objects (notice that 7+7+2+2+2 = 20), so there are


ways to do this.

To justify this, consider that we will arrange the n objects in a 
line.  Then the first box (or child) gets the first a_1 objects, the
second box gets the next a_2 objects, and so on.  There are n! ways to
arrange the objects in a line, but then we can rearrange the first a_1
objects without changing the outcome, and we can rearrange the next
a_2 objects, and so on.

Finally, just as the binomial coefficients (n-choose-r) come up in the
expansion of (x + y)^n, so too the multinomial coefficients come up in
the expansion of (x + y + z)^n (or with more variables).  For more
info on this, see

  Multinomial Coefficients

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
Associated Topics:
High School Permutations and Combinations

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