Final answer:
To find the number of ways the dogs can line up in front of the judges, we can use the concept of permutations. We need to calculate the number of ways to arrange the different breeds of dogs. To calculate the number of different ways for three dogs to win first, second, and third place, we can use the concept of combinations.
Step-by-step explanation:
To find the number of ways the dogs can line up in front of the judges, we can use the concept of permutations. Since dogs of the same breed are considered identical, we need to calculate the number of ways to arrange the different breeds of dogs.
First, we calculate the number of ways to arrange the Pomeranians, golden retrievers, Great Pyrenees, and English terriers separately. Since there are 4 Pomeranians, 5 golden retrievers, 2 Great Pyrenees, and 6 English terriers, we have:
Pomeranians: 4!
Golden Retrievers: 5!
Great Pyrenees: 2!
English Terriers: 6!
Then, we multiply these values together to get the total number of ways to arrange the dogs:
Total number of ways = 4! * 5! * 2! * 6!
To calculate the number of different ways for three dogs to win first, second, and third place, we can use the concept of combinations. We have a total of 17 dogs, so the number of combinations of 3 dogs is:
Number of combinations = 17 choose 3 = 17! / (3! * (17-3)!)
After calculating the above values, you will get the answer to the question.