37. Water Pollution An environmental agency is analyzing water samples from 80 lakes for pollution. Five of the lakes have dangerously high levels of dioxin. Six lakes are randomly selected from the sample. Use technology to find how many ways one polluted lake and five nonpolluted lakes can be chosen.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the number of ways to select 1 polluted lake (out of 5) and 5 non-polluted lakes (out of 75) from a total of 80 lakes. This is a combination problem because the order of selection does not matter.
2. Step 2: Recall the formula for combinations, which is given by: $C(n,r)=\frac{n}{!}$, where n is the total number of items, and r is the number of items to choose.
3. Step 3: Calculate the number of ways to choose 1 polluted lake from the 5 polluted lakes using the combination formula: $C(5,1)$. This represents the number of ways to select 1 polluted lake.
4. Step 4: Calculate the number of ways to choose 5 non-polluted lakes from the 75 non-polluted lakes using the combination formula: $C(75,5)$. This represents the number of ways to select 5 non-polluted lakes.
5. Step 5: Multiply the results from Step 3 and Step 4 to find the total number of ways to select 1 polluted lake and 5 non-polluted lakes. The final expression is: $C(5,1)\times C(75,5)$.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In this context, it helps determine the number of ways to select a specific number of items from a larger set, which is essential for calculating how many ways one polluted lake and five nonpolluted lakes can be chosen from the sample.

Probability

Probability is the measure of the likelihood that an event will occur. Understanding probability is crucial in this scenario as it allows the environmental agency to assess the chances of selecting a polluted lake among the total lakes sampled, which informs risk assessments related to water pollution.

Sampling Techniques

Sampling techniques refer to the methods used to select a subset of individuals or items from a larger population. In this case, the random selection of lakes is a sampling technique that ensures each lake has an equal chance of being chosen, which is important for the validity of the analysis regarding pollution levels.