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?
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To solve this problem, we can use the concept of combinations.
First, let's consider the flavors. Since any two ice cream cones in one order must differ by at least one flavor, we need to choose a different flavor for each pair of cones.
Out of the 30 flavors, we need to choose 2 flavors for each of the 12 two-scoop cones. This can be done through combinations, denoted as "n choose r". In this case, we have 30 flavors to choose from, and we need to choose 2 flavors. So, we have 30C2 choices for the first pair, 28C2 choices for the second pair, 26C2 choices for the third pair, and so on. This can be calculated as:
30C2 * 28C2 * 26C2 * ... * 4C2 * 2C2.
Now, let's consider the types of cones. Since any two ice cream cones in one order must also differ in the kind of cone, we need to choose a different kind of cone for each pair of cones.
Out of the 3 kinds of cones, we need to choose 2 kinds for each of the 12 two-scoop cones. Again, this can be done through combinations. We have 3 kinds of cones to choose from, and we need to choose 2 kinds. So, we have 3C2 choices for the first pair, 2C2 choices for the second pair, and so on. This can be calculated as:
3C2 * 2C2 * ... * 2C2.
Finally, to find the total number of ways to order a dozen two-scoop ice cream cones, we need to multiply the number of choices for flavors and cones:
(30C2 * 28C2 * 26C2 * ... * 4C2 * 2C2) * (3C2 * 2C2 * ... * 2C2).
Calculating this expression will give us the answer.