At a dog show there are 5 dogs competing for Best of Show. In how many orders can the judge pick out and rank the top 2 dogs?

To solve this problem, we need to use the concept of permutations.

Step 1: Understand the problem The problem is asking for the number of ways the judge can pick out and rank the top 2 dogs out of 5. This is a permutation problem because the order in which the dogs are picked matters (i.e., Dog A being first and Dog B being second is different from Dog B being first and Dog A being second).

Step 2: Apply the formula for permutations The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

Step 3: Plug in the numbers In this case, n = 5 (the total number of dogs) and r = 2 (the number of dogs to choose). So the calculation would be:

5P2 = 5! / (5-2)! = 54321 / 321 = 20

So, there are 20 different ways the judge can pick out and rank the top 2 dogs.