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?
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?
Solution
The problem you're describing is a classic example of a problem in combinatorics, specifically a problem of combinations with repetition.
This type of problem is solved using the formula for combinations with repetition, which is:
C(n + r - 1, r)
where n is the number of types of items (in this case, types of candies), and r is the number of items to choose (in this case, candies).
So, in this case, n = 5 (cherry, strawberry, orange, lemon, pineapple) and r = 27 (the number of candies to choose).
Plugging these values into the formula, we get:
C(5 + 27 - 1, 27) = C(31, 27)
This is a combination calculation, which can be calculated as:
C(31, 27) = 31! / [(31-27)! * 27!]
= 31! / (4! * 27!)
= (31 * 30 * 29 * 28) / (4 * 3 * 2 * 1)
= 27,405
So, there are 27,405 ways to choose 27 candies from 5 types of candies.
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