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Laci's wardrobe has 11 shirts of different colors. After separating his shirts into three piles of 2,3, and 6 shirts, in how many ways can Laci select 3 shirts to bring on a camping trip if he wants the selected shirts to come from at least two different piles?
There are 3 ways to choose 3 shirts from at least two different piles.
Choose 2 shirts from one pile and 1 shirt from another pile. There are 2 ways to choose the first pile, 3 ways to choose the second pile, and 2 ways to choose the shirt from the second pile. This gives us 12 ways to choose 2 shirts from one pile and 1 shirt from another pile.
Choose 1 shirt from each of two piles. There are 3 ways to choose the first pile, 2 ways to choose the second pile, and 3 ways to choose the third pile. This gives us 18 ways to choose 1 shirt from each of two piles.
Choose 1 shirt from each of three piles. There are 3 ways to choose the first pile, 2 ways to choose the second pile, and 1 way to choose the third pile. This gives us 6 ways to choose 1 shirt from each of three piles.
Adding these three ways together gives us 36 ways to choose 3 shirts from at least two different piles.
However, we need to subtract one way from this total, because we have counted the case where Laci chooses all three shirts from the pile of 6 shirts twice. This gives us a final answer of 35 ways.
