combination (tough)

mariah wrote:A professor will assign seven projects to three students. If two students each got 2 projects, and the other one got 3 projects, how many ways are possible?

oa is 630
my answer - 7C2*5C2*3C3=210
please explain.

7C3*4C2*2C2 + 7C2*5C3*2C2 + 7C2*5C2*3C3 = 630

To add a little explanation. Notice that the correct answer is 3 times the answer you got. That is becuase you chose to have the professor assign 2 projects to student A, then two projects to student B then the remaining 3 to Student C.

But there are two other ways the professor can assign the projects:

First 3 to Student A, then 2 to Student B then the remaining 2 to student C
and
First 2 to student A then 3 to student B and the remaining 2 to student C.

This gives us the 3 equations to add together.

7C2 * 5C2 *3C3

7C3 * 4C2 *2C2

7C2 * 5C3 * 2C2

Each one of them gives 210 combinations for a total of630 combinations.

-Carrie