You may leave factorials and binomial coefficients unevaluat

You may leave factorials and binomial coefficients unevaluated in this question. A store sells 4 types of hard candies: cherry, strawberry, orange and lemon. How many ways are there to choose: (a) 20 candies with at least a piece of each flavor? (b) 20 candies with at least 4 cherry, 5 strawberry, 2 lemon and any amount (including zero) of orange?

Solution

a)

If the first four are fixed to one of each flavor, then there are 4^16 = 4294967296 ways to choose the other 16 any way we want.

Thus, there are 4294967296 ways. [ANSWER]

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b)

If we set the first eleven to be 4 cherry, 5 strawberry, and 2 lemon, then there are 4^9 = 262144 ways to choose the other nine candies any way we want.

Thus, there are 262144 ways. [ANSWER]