Suppose there are 5 Dachshunds, 4 Pomeranians, and 6 Golden Retrievers at a veterinarian's office. The vet that owns the practice needs to select 10 dogs out of the group to photograph for a lecture he is giving about pet dental care. How many different ways can the 10 dogs be chosen?
Question
Suppose there are 5 Dachshunds, 4 Pomeranians, and 6 Golden Retrievers at a veterinarian's office. The vet that owns the practice needs to select 10 dogs out of the group to photograph for a lecture he is giving about pet dental care. How many different ways can the 10 dogs be chosen?
Solution
This problem can be solved using the concept of combinations in statistics.
First, let's understand the total number of dogs. We have 5 Dachshunds, 4 Pomeranians, and 6 Golden Retrievers, which gives us a total of 15 dogs.
The vet needs to select 10 dogs out of these 15. The order in which the dogs are selected does not matter, so we use the combination formula:
C(n, r) = n! / [r!(n-r)!]
where:
- n is the total number of items,
- r is the number of items to choose,
- "!" denotes factorial, which means multiplying all positive integers up to that number.
So, in this case, n = 15 (total dogs) and r = 10 (dogs to be chosen).
Substituting these values into the formula, we get:
C(15, 10) = 15! / [10!(15-10)!]
Calculating the factorials and simplifying, we find the number of ways the vet can choose 10 dogs out of 15 for the photograph.
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