Question Video: Applications of the Counting Principle (Product Rule) And Combinations | Nagwa Question Video: Applications of the Counting Principle (Product Rule) And Combinations | Nagwa

Question Video: Applications of the Counting Principle (Product Rule) And Combinations Mathematics • Third Year of Secondary School

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A teacher must pick a committee of 6 students consisting of 4 boys and 2 girls. In how many ways can the committee be formed if the teacher has 8 boys and 7 girls to choose from?

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Video Transcript

A teacher must pick a committee of six students consisting of four boys and two girls. In how many ways can the committee be formed if the teacher has eight boys and seven girls to choose from?

In this question, we are told that a teacher needs to select a committee consisting of six total students, four boys and two girls. We want to determine the total number of possible committees if the teacher has eight boys and seven girls to choose from. To do this, we can start by recalling that 𝑛 choose π‘˜ is the number of ways of choosing π‘˜ objects from 𝑛 distinct objects. We can use this to find the number of ways of choosing π‘˜ things from 𝑛 things, provided the order we choose does not matter.

In this case, we know that the order we choose the committee does not matter. The only differences are who is chosen for the committee. We can also recall that 𝑛 choose π‘˜ is equal to 𝑛 factorial over π‘˜ factorial times 𝑛 minus π‘˜ factorial. Let’s use this result to determine the number of ways of choosing four boys from the eight candidates. We have 𝑛 equals eight and π‘˜ equals four, so this is given by eight choose four. Substituting these values into the formula yields eight factorial over four factorial times eight minus four factorial. We can simplify this expression to obtain eight factorial over four factorial times four factorial.

We can simplify further by recalling that the factorial of a number is the product of the positive integers less than or equal to the number. In particular, eight factorial can be written as eight times seven times six times five times four factorial. Canceling the shared factor of four factorial in the numerator and the denominator and then evaluating gives us that there are 70 ways of choosing four boys from the eight candidates who are boys.

We can follow the same process to determine the number of ways of choosing two girls from the seven candidates who are girls. We substitute 𝑛 equals seven and π‘˜ equals two into the formula to get seven factorial over two factorial times seven minus two factorial. We can then simplify and evaluate this expression in the same way. We have that seven factorial over five factorial is equal to seven times six. Then we divide this by two factorial to obtain 21. Hence, there are 21 ways of choosing two girls from seven to be on the committee.

We now need to combine these results to find the total number of ways the teacher can select the entire committee. To do this, we first note that choosing the boys and choosing the girls for the committee are independent. The choices of one group will not affect the other. Since the events are independent, we can use the fundamental counting principle to find the number of ways of choosing the committee. We recall that this tells us that we take the product of the number of ways of choosing from each set to find the total number of possibilities. Hence, the number of different committees that can be formed by the teacher is 70 times 21, which is equal to 1,470.

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