A representative of Environmental Protection Agency wants to select samples from 10 landfills. The director

has 15 landfills from which she can collect samples. How many different samples are possible?

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.

Understand how to calculate combinations in probability theory. Solve the problem of selecting samples from 15 landfills where 10 need to be chosen. Learn the combinations formula and step-by-step calculation process. Suitable for high school students studying probability and combinatorics.

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