How many ways can eight identical svarves be placed into four disingusihable boxes( boxes you can tell apart), if each box must contain at least one scarf?

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- Jun 18th 2008, 05:32 AMsantatrueCounting!
How many ways can eight identical svarves be placed into four disingusihable boxes( boxes you can tell apart), if each box must contain at least one scarf?

- Jun 18th 2008, 07:11 AMSoroban
Hello, santatrue!

Quote:

How many ways can eight identical scarves be placed into four distinguishable boxes

( boxes you can tell apart), if each box must contain at least one scarf?

*list*the possiblities . . . *blush*

I found five scenarios.

$\displaystyle 5,1,1,1$ . . . and there are $\displaystyle 4$ ways.

$\displaystyle 4,2,1,1$ . . . and there are $\displaystyle \frac{4!}{2!} = 12$ ways.

$\displaystyle 3,3,1,1$ . . . and there are $\displaystyle \frac{4!}{2!2!} = 6$ ways.

$\displaystyle 3,2,2,1$ . . . and there are $\displaystyle \frac{4!}{2!} = 12 $ ways.

$\displaystyle 2,2,2,2$ . . . and there is $\displaystyle 1$ way.

Therefore, there are: .$\displaystyle 4 + 12+6+12+1 \:=\:\boxed{35\text{ ways}}$

- Jun 18th 2008, 07:11 AMPlato