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.  How many groups of 7 can be chosen to work on the project?

Linda,

Let's use  nCr 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  nCr.
  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

     2C0n-2Cr + 2C1n-2Cr-1 + 2C2n-2Cr-2.

Thus from the above

 nCr =  2C0n-2Cr +  2C1n-2Cr-1 +  2C2n-2Cr-2 as we have answered the problem of choosing r people correctly in two ways.

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

 2C010C7 +  2C110C6, i.e., you pick 7 others or pick 1 of your special pair and 6 others.

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

 2C010C7 +  2C210C5, i.e., you pick 7 others or pick both of your special pair and 5 others.

Penny
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