A store is selling 5 types of hard candies: cherry, strawberry, orange, lemon and pineapple. How many ways are there to choose:(a) 27 candies?Answer = (b) 27 candies with at least a piece of each flavor?Answer = (b) 27 candies with at least 2 cherry and at least 4 lemon?Answer =
Question
A store is selling 5 types of hard candies: cherry, strawberry, orange, lemon and pineapple. How many ways are there to choose:(a) 27 candies?Answer = (b) 27 candies with at least a piece of each flavor?Answer = (b) 27 candies with at least 2 cherry and at least 4 lemon?Answer =
Solution
Sure, I can help with that. This is a problem of combinations with repetition, also known as multiset coefficients or "combinations with unlimited supply".
(a) The number of ways to choose 27 candies out of 5 types is given by the formula C(n + r - 1, r), where n is the number of types (5 in this case) and r is the number of candies (27 in this case). So, the answer is C(5 + 27 - 1, 27) = C(31, 27) = 31,381.
(b) If we want at least one of each flavor, we can first give one candy of each flavor. That leaves us with 27 - 5 = 22 candies to distribute among 5 flavors. Using the same formula as above, we get C(22 + 5 - 1, 22) = C(26, 22) = 14,950.
(c) If we want at least 2 cherry and at least 4 lemon, we can first give 2 cherry and 4 lemon. That leaves us with 27 - 2 - 4 = 21 candies to distribute among 5 flavors. Using the same formula as above, we get C(21 + 5 - 1, 21) = C(25, 21) = 4,950.
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