Problem number 21. Jane has some drawings of parrots. She wants to color only the head, tail, and wings of each parrot, red, blue, or green, so that all three colors are used on each picture. She colors one parrot's head red, its wings green, and its tail blue. How many more parrots can she color so that all the parrots are colored differently? A1, B2, C4, D5, E9. There are three regions on each parrot that Jane is going to color, the head, the tail, and the wings, and she wants to use three colors, red, blue, and green. What we need to find is how many different combinations there are that allow Jane to color the parrot the way she wants to. So here are some more pictures of parrots. Let's start with the combination that Jane used. She colored the head red, the wings green, and the tail blue. Let's start with a red head and see how many more ways we can do it. We have a red head. We only have two colors left, green and blue, and since we already have a picture with green wings, we're going to color one with blue wings, and the tail will be green. If we were to color another head red, we would have to repeat the combination, making the wings green and the tail blue, or the other way around. There's no other way to do this. So we only have two choices so far. Now let's make the head. We can make the wings red and the tail green, or we can make the wings green and the tail red. Those are our only two choices in this case. Likewise, if we make the head green, we can make the wings red and the tail blue, or the other way around. The wings can be blue and the tail red. So we have two more choices. We've used up all the different colors for the head and found all the different ways to then color the wings and the tail. We have six choices total as to how to color this parrot. Now since Jane already made one of the pictures, she is left with six minus one, or five different ways to color the parrot. So the answer is D, five. The mathematical way to find this is to think about how many choices for the first element we have. So if we start with the head, we can color it three different ways. Then for the tail, we are left with only two color choices, because we've used one of them up for the head. And for the wings, we are left with whichever color we didn't use, so we don't really have a choice. You multiply those numbers together, and you get the number of possible combinations, which is six. And again, you would subtract the one that you already colored, and you're left with five minus one. The answer is D, five.

parrot coloring

color combinations

unique arrangements

head tail wings

three colors