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SOLUTION to Problem 2 As suggested by the hint, we will count the number of ways to choose four balls from a collection of n+1 balls. There are C(n+1,4) ways to do so, by definition. Now imagine that one of the balls is hot pink. If this ball is to be one of the chosen four, then we need only choose three of the remaining n balls to accompany it, which can be done in C(n,3) ways. The only other possibility is that the hot pink ball is not one of the chosen four, meaning that we choose all four from the remaining n balls. Of course, this can be done in C(n,4) ways. Since we are counting the same thing either way, both approaches must yield the same final answer, which proves the claim. |