There are 1,771 ways to choose 24 croissants with at least five chocolate and at least three almond croissants from the shop.
How to solve
You must select 24 croissants, ensuring at least five chocolates and three almonds. Total options: 6 types, requiring minimum quantities of chocolates and almonds.
Separate the guaranteed choices: First, choose the minimum number of chocolate and almond croissants: 5 chocolate and 3 almond. This leaves 24 - 5 - 3 = 16 croissants to choose from.
Treat the rest as combinations: Now, you need to choose 16 croissants from the remaining 4 types (plain, cherry, apple, and broccoli). This is a combination problem with repetition allowed (you can choose the same type multiple times) since you can still have more chocolate or almond croissants.
Formula for combinations with repetition: The formula for combinations with repetition is n + r - 1 choose r. In this case, n = 4 (remaining croissant types), r = 16 (croissants to choose), so we have:
4 + 16 - 1 choose 16 = 19 choose 16 = 1,771
Therefore, there are 1,771 ways to choose 24 croissants with at least five chocolate and at least three almond croissants from the shop.
The Complete Question
A croissant shop offers a variety of delicious options: plain, cherry, chocolate, almond, apple, and even broccoli (for the adventurous eaters!). You want to purchase 24 croissants, ensuring you get at least 5 chocolate and at least 3 almond croissants. In how many ways can you fulfill your croissant craving?