Partial Permutations or Combinations Without Replacement

[Metronome]

If we sample [MATH]r[/MATH] balls without replacement from an urn containing [MATH]n[/MATH] balls, where [MATH]b[/MATH] of the [MATH]n[/MATH] are blue, [MATH]g[/MATH] of the [MATH]n[/MATH] are green, [MATH]y[/MATH] of the [MATH]n[/MATH] are yellow, etc., for some finite number of colors, what formulas could be used to find the number of possibilities with and without order significance?

 

[Steven G]

You know by now that we do not solve any problems for students on this forum.

We have a show no work, get no help policy.

So please tell us where you are stuck and show us your work.

Hint: If this was my problem I would thing about the number of ways I could add up positive integers which sum to n.

 

[pka]

If we sample [MATH]r[/MATH] balls without replacement from an urn containing [MATH]n[/MATH] balls, where [MATH]b[/MATH] of the [MATH]n[/MATH] are blue, [MATH]g[/MATH] of the [MATH]n[/MATH] are green, [MATH]y[/MATH] of the [MATH]n[/MATH] are yellow, etc., for some finite number of colors, what formulas could be used to find the number of possibilities with and without order significance?

Let \(r=r_b+r_g+r_y\) where the subscript denotes the colour so \(0\le r_b\le b,~0\le r_g\le g,~\&~0\le r_y\le y\)
Note that \( b+g+y= n\). The number of ways to choose \(r_b\) blue balls, \(r_g\) green balls.\(~\&~r_y\) yellow balls is
(b choose \(r_b)~\)(g choose \(r_g\)) (y choose \(r_y)\)