6. A shipment of 250 netbooks contains 3 defective units. Dete...

Question

6. A shipment of 250 netbooks contains 3 defective units. Determine how many ways a vending company can buy three of these units and receive

c. at least one good unit.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A warehouse has 180 smartphones, 4 of which are defective. How many ways can a retailer select 3 smartphones and receive at least 1 that is not defective? Awesome. So it appears for this particular prom, we need to take the conditions that are set to us by the prom itself and determine how many ways a retailer can select 3 random random smartphones. And receive at least one that is not defective. Awesome. So with that in mind, now that we know we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 955,836. B is 955,856. C is 5,735,136. D is 5,735,160. Awesome. So first off, let us recall a note that when faced with a selection problem involving at least one condition, such as at least one is not defective, which is the case of this problem. It is often easiest to recall and use the complement principle, and as we should recall, the complement principle involves calculating the total number of ways to make the selection with no restrictions. And subtracting the number of ways in which the unwanted condition occurs, which in this case for this problem, all the selected items are defective. So, Our first major step to solve for this particular problem is we need to determine the total number of ways to choose 3 smartphones from the 180 total. So the number of ways to choose 3 from 180 is C of 1803, which is another way of saying this is 180 choose 3, so using the combination formula, as we should recall, we can write that 180 factoral. So we could say that C of 180.3 is equal to 180 factoral, divided by 180 minus 3 multiplied by factoral, multiplied by 3 factoral. So when we plug that into our calculator, we will find that it's equal to 955,860. Fantastic. So now, Our second step, and once again for step one, we had to recall and use the combination formula to solve for C of 180.3 or you might hear it as C of 180, choose 3. Awesome. So our second step now is we need to determine the number of ways to choose 3 smartphones that are all defective. So as we should recall, there are only 4 defective smartphones total. So the number of ways to choose 3 defective phones is C4 chooses 3 or 4.3. is equal to 4 factoral divided by 4 minus 3 multiplied by factoral, multiplied by 3 factoral, which is equal to 4. So now, Our 3rd step is we need to determine the number of ways to choose 3 smartphones with at least 1 of them not being defective. So, at least one not defective. So I I could say at least. One Not defective, which I'll call A1 and D for short, at least 1 not defective, is equal to the total number of ways minus the all defective. Which I'll call all death for short, which will be equal to, when we plug in our known variables, 955,860 minus 4, which is gonna be equal to 955,856 and that is our final answer. Hooray, we did it. So therefore, without a doubt, there are 955,856 ways for a retailer to select 3 smartphones and receive at least one that is not defective. And this corresponds with multiple choice answer letter B, which once again is 955,856. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

6. A shipment of 250 netbooks contains 3 defective units. Determine how many ways a vending company can buy three of these units and receive
c. at least one good unit.

Key Concepts

Combinatorics

Complementary Counting

Probability

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