Sara's questions at Yahoo Answers regarding permutations and combinations

[MarkFL]

Here are the questions:

How to solve this question about permutations? How do you get to this answer? 10 pts for best answer! :)?

Please show how you arrived at the answer (I want a detailed process explanation!)
Here's the answer to the first question: 8.88 x 10^ (19)
But how do you arrive at this answer?

Question:
In how many ways can 30 teachers be assigned to 6 schools, with each school receiving an equal number of teachers?And if you're up to the challenge ... :
Q: In how many ways can 12 jurors and 3 alternate jurors be selected from a group of 25 prospective jurors?

Thanks a bunch!

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello Sara,

1.) Obviously we are going to have to send 5 teachers to each of the 6 schools. There are $30!$ ways to order the 30 teachers, but we have to account for the fact that for each of the 6 groups of 5 teachers, there are 5! ways to order them, and since order does not matter, we find the number of ways $N$ to do this is:

\(\displaystyle N=\frac{30!}{(5!)^6}=88832646059788350720\approx8.88\times10^{19}\)

2.) We need to compute the number of ways to choose 15 from 25, and multiply this with the number of ways to choose 3 from 15, hence:

\(\displaystyle N={25 \choose 15}{15 \choose 3}=1487285800\)