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?