In this scenario, the Environmental Protection Agency's representative wants to select samples from 10 landfills out of a possible 15. Since the order in which the landfills are selected does not matter, this is a combination problem.

The formula for combinations is:

$C(n, r) = \frac{n!}{r!(n-r)!}$

Where:

• $n$ is the total number of items to choose from (15 landfills),
• $r$ is the number of items to choose (10 landfills).

Substituting the values:

$C(15, 10) = \frac{15!}{10! \times (15-10)!} = \frac{15!}{10! \times 5!}$

Let's calculate this step by step:

$15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10! = 360360 \times 10!$ $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800$ $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ $C(15, 10) = \frac{360360 \times 3628800}{3628800 \times 120} = \frac{360360}{120} = 3003$

Thus, there are 3,003 different possible samples.

Would you like any further details or explanations?

Here are some related questions to consider:

1. How would the solution change if the representative only needed to select 5 landfills?
2. What if the order of selection mattered? How many different selections would be possible then?
3. How does this problem differ from a permutation problem?
4. What if the representative wanted to select all 15 landfills?
5. What is the probability of selecting a specific set of 10 landfills?

Tip: Combinations are used when the order of selection does not matter; permutations are used when the order does matter.