I was just helping a student with his AoPS homework, when I came across the following related problems:
Eight people, including Fred, are in a club. They decide to form a 3 person committee. How many possible committees can be formed?
So we are choosing 3 people out of 8 or .
How many possible committees include Fred?
Since Fred is taking one committee seat, that means we need to choose 2 more people from the remaining seven, or .
How many possible committees do not include Fred?
Since we can’t choose Fred, we need to choose 3 members from the remaining 7 or .
Since the total number of committees is equal to the number of of committees with Fred plus the number of committees without Fred, then we can say
Generally we call this Pascal’s Rule: .