The top 10 graduates from Law School with Ethics were offere

The top 10 graduates from Law School with Ethics were offered positions with the three largest law firms in a certain city: the largest firm offered 4 positions, the second and the third largest firms each offered three positions. Suppose the 10 graduates were randomly assigned to these 10 positions. What is the probability that the top three students end up working for the third largest firm? In the 10 top graduates, there axe three sisters. What is the probability that the three sisters end up working for the same firm?

Solution

a)

There are, by permutation of like objects, 10!/[4!3!3!] = 4200 ways to distribute them in the three companies.

Setting the top 3 people on the last company, then there are 7!/(4!3!) = 35 ways to do that.

Thus, the probability that the top 3 end up working on the third firm is

P = 35/4200 = 0.008333333 [answer]

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b)

There are 3 ways to do this:

1. The sisters work for firm 1.
2. The sisters work for firm 2.
3. The sisters work for firm 3.

For these cases,

1. Since we fix the first three, there are 7!/(1!3!3!) = 140 ways to fix the other 7.

2. This time, there are 7!/(4!3!) = 35 ways to arrange the other 7.

3. Again, there are 7!/(4!3!) = 35 ways to arrange the other 7.

Therefore, a total of 140+35+35 = 210 ways the sisters can work for the same company.

Thus,

P = 210/4200 = 0.05 [answer]