1. Two headed robots

- Great job completing the last tutorial! You made both the director and the producer very happy. Sometimes our directors change their minds when they see their drawings translated to 3D computer graphics. In fact, sometimes our directors change their minds a lot, and we need to be able to adapt to those changes. So let's say we have a director that wants a robot with two heads, but each head has to be different. That is, the director doesn't want a robot with two copies of the same head. Now a robot consists of two heads, a body, and arms. We'd like to know how many of those we can have if we have H heads, B bodies, and A arms. We saw that by using a Tree Diagram we could multiply the number of each kind of part together to get the total number of combinations. Well, we have H choices for the first head, but since the second head has to be different, we only have H minus one choices for the second head. As before, we have B choices for bodies and A choices for arms, for a total of H times H minus one times B times A combinations. So it's all about multiplying together all the choices you have at each stage. Mathematicians make up words for everything. They use the word combinatorics to refer to this type of counting. The next exercise will give you a chance to practice with other combinations.