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.
To find the number of ways to order a dozen two-scoop ice cream cones with the given conditions, we can break down the problem into several steps:
Step 1: Choose the flavor combinations
In each ice cream cone, we can choose any two flavors from the 30 available flavors. Since order does not matter, we can use combinations. The number of ways to choose 2 flavors out of 30 is calculated using the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of flavors and k is the number of flavors to be chosen.
C(30, 2) = 30! / (2!(30-2)!)
= 30! / (2!28!)
= (30 * 29) / (2 * 1)
= 30 * 29 / 2
= 435
Therefore, we have 435 different combinations of flavors for each ice cream cone.
Step 2: Choose the kind of cone combinations
Similarly, we have 3 different kinds of cones to choose from. Using the same combination formula, we can calculate the number of ways to choose 2 kinds of cones out of 3.
C(3, 2) = 3! / (2!(3-2)!)
= 3! / (2!1!)
= 3 / 2
= 3
So, there are 3 different combinations of kinds of cones for each ice cream cone.
Step 3: Multiply the possibilities
Since all the ice cream cones need to differ either by flavor or kinds of cones, we need to multiply the number of possibilities from step 1 and step 2.
Total possibilities = 435 * 3
= 1305
Therefore, there are 1305 ways to order a dozen two-scoop ice cream cones with the given conditions.