I have 6 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?
I can use the multiplication principle: if I have a choices for one thing, and b choices for another thing, and they are independent, then I have a*b choices for both things. So, how many choices do I have for each item? For shirts, I have 6 choices, regardless of the color. For pants, I have 4 choices, regardless of the color. For hats, I have 6 choices, regardless of the color. So, if I didn't care about the color matching, I would have 6*4*6 = 144 choices for outfits. But, I do care about the color matching, so I need to subtract the cases where all three items are the same color. How many cases are there where all three items are the same color? Well, I only have 4 colors that are common to all three items: tan, black, blue, and gray. So, for each of those colors, I have one choice for each item, so 1*1*1 = 1 choice for an outfit of that color. So, I have 4 cases where all three items are the same color. Therefore, I need to subtract 4 from the total number of choices. So, the final answer is 144 - 4 = 140 choices for outfits. 