Question

an environmental agency is analyzing water samples from 78 lakes for pollution. five of the lakes have dangerously high levels of dioxin. six of the lakes are randomly selected from the sample. using technology, how many ways could one polluted lake and five non - polluted lakes be chosen?

therefore, there are
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the number of ways to choose 1 polluted lake

The number of polluted lakes is 5. The number of ways to choose 1 polluted lake is given by the combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n = 5\) and \(k=1\). So \(C(5,1)=\frac{5!}{1!(5 - 1)!}=\frac{5!}{1!4!}=5\)

Step2: Calculate the number of non - polluted lakes

The total number of lakes is 78, and the number of polluted lakes is 5. So the number of non - polluted lakes is \(78-5 = 73\)

Step3: Calculate the number of ways to choose 5 non - polluted lakes

We use the combination formula with \(n = 73\) and \(k = 5\). \(C(73,5)=\frac{73!}{5!(73 - 5)!}=\frac{73\times72\times71\times70\times69\times68!}{5\times4\times3\times2\times1\times68!}=13852818\)

Step4: Calculate the total number of ways to choose 1 polluted and 5 non - polluted lakes

By the multiplication principle, we multiply the number of ways of choosing a polluted lake and the number of ways of choosing non - polluted lakes. So the total number of ways is \(C(5,1)\times C(73,5)=5\times13852818 = 69264090\)

Answer:

69264090