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I know the bottom should be n+p choose n but I'm not sure what the numerator should be, any ideas would be great.

I was thinking n+p choose m since that will give the different ways that the red seats could be chosen, times n+p choose n-m which gives the choices that the non-red seats could be chosen.

Or: [tex] \frac{ C^{n+p} _{m} C^{n+p} _{n-m} } { C^{n+p} _{n} } [/tex]

Does this logic make sense?