4. Casting problem

earlier you created a small crowd of robots by choosing just a few of the possible robot combinations let's back up a bit and go back to the example where robots consisted of one head and one body if you have two heads and two bodies then you can make 2 times 2 or 4 different robots that's like having 4 possible actors to choose from but you only need a cast of 3 in your movie let's give our actors names Alice Bob Carol and Dave or a B C and D for short that leads to an interesting question how many different casts of three actors can you make when you have four actors to choose from remember that to figure out how many ways there are to combine our actors we need to multiply our choices at each stage in this case we have four choices for our first actor three for the second and two for the third so it seems like we have four times three times two or twenty four possible casts let's list out all 24 combinations the first combination ABC means we selected Alice then Bob and then Carol but there's a subtlety lurking in there there aren't twenty-four different casts to see where the subtlety is notice that the second combination a C B means we selected Alice then Carol and then Bob so the first two combinations use the same actors just in a different order the same is true for the other combinations in the first box it's the same cast just the order that we picked them from was different in other words all the combinations in the first box should be counted as just one cast similarly the second box is a cast consisting of Alice Bob and Dave the total number of casts is therefore the number of boxes so how many boxes are there since there are six combinations in each box there must be 24 divided by six or four boxes so there are four cast but why is it that each group contains exactly six sequences why not three or four or any other number the next interactive will help you visualize this problem using a few other examples have fun