Video: Application of the Counting Principle (Product Rule)

A building has 5 doors which are numbered as 1, 2, 3, 4, 5. Determine the number of ways a person can enter and then leave the building if they cannot use the same door twice.

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Video Transcript

A building has five doors which are numbered as one, two, three, four, five. Determine the number of ways a person can enter and then leave the building if they cannot use the same door twice.

Let’s try to visualize this. Our building has five doors, and let’s label them one, two, three, four, five as it tells us to. Let’s imagine we have someone looking to enter the building. They have five possible ways to do so. But let’s imagine for sake of argument that they’re going to choose door number two. Once inside the building, we’re told they cannot use the same door twice, and so we cut off door two as an exit. Looking around, we now see that there are one, two, three, four possible ways for that person to exit the building. They may, for example, choose door four. There are therefore five possible ways to enter the building. But once we are into the building, there are only four possible ways to get out.

The product rule for counting or the counting principle tells us that the total number of ways a person can enter and then leave the building given these restrictions is the product of these. It’s five times four, which is equal to 20. There are 20 possible ways then that the person can enter and then leave the building given that they can’t use the same door twice.

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