Probability/Counting Rules Question

[skhan]

Hello:
I was having trouble answering these two probability questions, so assistance from anyone would be much appreciated.

A project director runs a staff consisting of 6 scientists and 3 lab technicians. Three new projects have to be worked on and the director decides to assign 4 of her staff to the first project, 3 to the second project and 2 to the third project. In how many ways can this be accomplished if:

a) Of the 4 people assigned to the first project, at least 3 are scientists? ANS: 750

So I tried this problem, and i don't get 750 which is pretty frustrating...
Heres what I did:

Let's say there are 3 scientists on the 1st project:

1st group 2nd group 3rd group
------------ ----------- ------------

3S 1LT 1S 2LT 2S 0LT
3S 1LT 2S 1LT 1S 1LT
3S 1LT 3S 0LT 0S 2LT

Let's say there are 4 scientists on 1st project:

1st group 2nd group 3rd group
---------- --------- ---------
4S 0LT 2S 1LT 0S 2LT
4S 0LT 1S 2LT 1S 1LT

Counting up all these (omitting combinations which equal 1):

C(6,3)C(3,1)C(3,1)+C(6,3)C(3,1)C(3,2)C(2,1)+
C(6,3)C(3,1)+C(6,4)C(3,1)+C(6,4)C(2,1)C(3,2)
=180+360+60+45+90=735

...help

 

[Nimz]

Looks to me like you omitted one case for when there are 4 scientists on the 1st project. There isn't any condition that says there has to be a scientist on the 2nd project, is there?

By the way, you said you had trouble with two questions?

 

[tacman]

It looks like you have the right idea, but there are a few mistakes in your calculations. Here is a step-by-step explanation of how to solve this problem:

1. First, let's break down the problem into smaller parts. We need to assign 4 people to the first project, 3 people to the second project, and 2 people to the third project. We can do this by finding the number of ways to choose 4 people from 6 scientists, 3 people from 6 scientists, and 2 people from 9 total staff members.

2. To find the number of ways to choose 4 people from 6 scientists, we use the combination formula: C(6,4) = 15. This means there are 15 ways to choose 4 scientists from a group of 6.

3. Similarly, there are C(6,3) = 20 ways to choose 3 scientists from a group of 6, and C(9,2) = 36 ways to choose 2 people from a group of 9.

4. Now, we need to consider the conditions of the problem. We know that at least 3 of the 4 people assigned to the first project must be scientists. This means that we have to subtract the cases where only 2 or 1 scientists are assigned to the first project.

5. There are C(6,2) = 15 ways to choose 2 scientists from a group of 6, and C(6,1) = 6 ways to choose 1 scientist from a group of 6. So, we need to subtract 15+6 = 21 from the total number of ways to choose 4 people for the first project.

6. Now, we can calculate the total number of ways to assign people to the projects:

15 x 20 x 36 = 10,800 total ways to assign people

But we need to subtract the cases where there are not enough scientists for the first project. So, we subtract 21 from this total, giving us a final answer of 10,779 ways to assign people to the projects.

I'm not sure where the answer of 750 came from, as that seems to be a different problem. I hope this explanation helps!