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 different ways to select the lakes. (round to four decimal places as needed.)

Explanation:

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

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

Step2: Calculate the number of ways to choose 5 non - polluted lakes

There are \(78-5 = 73\) non - polluted lakes and \(k = 5\) to be chosen. So \(C(73,5)=\frac{73!}{5!(73 - 5)!}=\frac{73\times72\times71\times70\times69\times68!}{5!\times68!}=\frac{73\times72\times71\times70\times69}{5\times4\times3\times2\times1}=13977612\)

Step3: Calculate the total number of ways using the multiplication principle

By the multiplication principle (if one event can occur in \(m\) ways and another independent event can occur in \(n\) ways, then the two events together can occur in \(m\times n\) ways), the total number of ways to choose 1 polluted and 5 non - polluted lakes is \(C(5,1)\times C(73,5)\)

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Answer:

\(69888060\)