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

• indigojoker
In summary, the question asks for the probability of all m red seats being occupied in a classroom with n+p total seats. One approach is to use the combination formula, with the numerator being the number of ways to choose m red seats and n-m non-red seats, and the denominator being the total number of ways to choose n people from n+p seats. Another approach is to first allocate the m red seats and then choose the remaining n-m people from the remaining n+p-m 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?

indigojoker said:
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.

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

indigojoker said:
then:
$$\frac{ C^{n+p} _{m} C^{n+p-m} _{n-m} } { C^{n+p} _{n} }$$?
EDIT: my error; see the other post.

Last edited:

## 1. What is the probability that all n people will have a seat in a classroom with n+p seats?

The probability can be calculated using the formula P(all n people have a seat) = (n+p)!/(p!*(n-1)!*n^p).

## 2. How does the number of seats in the classroom affect the probability of all n people having a seat?

The probability decreases as the number of seats (p) increases. This is because there are more possible combinations for people to sit in, making it less likely for all n people to have a seat.

## 3. Is it possible for more than n people to have a seat in a classroom with n+p seats?

Yes, it is possible for more than n people to have a seat. This can happen if some people choose to share seats or if the classroom has additional chairs or space for people to sit.

## 4. What is the maximum number of people that can have a seat in a classroom with n+p seats?

The maximum number of people that can have a seat is n+p, assuming that everyone chooses to sit down and there are no additional chairs or space in the classroom.

## 5. How does the arrangement of seats in the classroom affect the probability of all n people having a seat?

The arrangement of seats does not affect the probability. As long as there are enough seats for all n people to sit in, the probability remains the same. This is because the probability is based on the total number of possible combinations, not the specific arrangement of seats.

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