Suppose there are 19 marbles in a bag: 5 red, 6 green, and 8 blue ones. How many ways can 5 marbles be selected if exactly 2 marbles are red and 3 are another color?
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This is a simple problem, and you don't need inclusion-exclusion, removing repetitions, and the like.
# of ways = (ways to select 2 red)*(ways to select any 3 of other colors)