N people sit down at random a classroom containing n+p seats

[indigojoker]

n people sit down at random a classroom containing n+p seats. There are m red seats (m<=n) in the classroom, what is the probability that all red seats will be occupied?

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: $$\frac{ C^{n+p} _{m} C^{n+p} _{n-m} } { C^{n+p} _{n} }$$

Does this logic make sense?

 

[Unco]

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.

Close, but after having allocated m people amongst the n+p seats there are now only n+p-m seats to allocate the remaining n-m people.

 

[indigojoker]

then:
$$\frac{ C^{n+p} _{m} C^{n+p-m} _{n-m} } { C^{n+p} _{n} }$$?

 

[Unco]

then:
$$\frac{ C^{n+p} _{m} C^{n+p-m} _{n-m} } { C^{n+p} _{n} }$$?

EDIT: my error; see the other post.