Name: Linda

Level: secondary
Asking: student

There are 12 people who can be chosen for a project.

Q1. Suppose that 2 people refuse to work together. How many groups of 7 can be chosen to work on a project?

Q2. Suppose that 2 people insist on working together or neither will work on the project. ÊHow many groups of 7 can be chosen to work on the project?

Linda,

Let's use  _nC_r for the number of ways of choosing r of n distinct objects. In general if you have n people, of which 2 are special and n-2 are not (as in your case) , and want to choose r of them, when thinking of how many ways you can select the r break the answer down into cases.

1. the number of way to pick r from n is  _nC_r.
   
2. in our selection we must do one of the following: pick 0 from the special
   
3. pick 1 of them, pick both of them. Thus the number of ways to pick r is
   
    ₂C₀* _n-2C_r + ₂C₁* _n-2C_r-1 + ₂C₂* _n-2C_r-2.

Thus from the above

In your Q1 you want all choices not involving the special two so the answer is

In your Q2 you want all choices with not exactly 1 of the special two so the answer is