An ice cream store sells 30 different flavours of ice cream and it offers a choice of 3 different kinds of cones. In how many ways can we order a dozen two-scoop ice cream cones if any two of them in one order must differ at least by a flavor or by the kinds of cones?
Thanks for your help.
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To solve this problem, we need to consider the number of ways we can choose the flavors and kinds of cones for each order.
Let's break down the problem:
1. Choosing the flavors: We have 30 different flavors, and we need to choose 2 scoops for each cone. Since any two scoops in one order must differ by flavor, we can choose the first scoop in 30 ways and the second scoop in 29 ways (since it must differ from the first scoop). Therefore, there are 30 * 29 = 870 ways to choose the flavors for each order.
2. Choosing the kinds of cones: We have 3 different kinds of cones, and we need to choose one kind for each cone in the order. Since any two cones in one order must differ by the kind of cone, we have 3 choices for each cone. Therefore, there are 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 3^12 = 531,441 ways to choose the kinds of cones for each order.
3. Combining the choices: To find the total number of ways to order a dozen two-scoop ice cream cones, we multiply the number of choices for flavors by the number of choices for kinds of cones for each order. Therefore, the total number of ways to order is 870 * 531,441 = 462,682,870.
So, there are 462,682,870 ways to order a dozen two-scoop ice cream cones, where any two of them in one order differ by flavor or kind of cone.
I hope this helps! Let me know if you have any further questions.