How Do You Calculate Samples Containing Defective Widgets?

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To calculate how many samples of size 5 contain at least one defective widget from a set of 100, where 3 are defective, first determine the total combinations of samples, which is 75,287,520. The probability that a sample of 5 contains no broken widgets can be calculated using the formula (97*96*95*94*93)/(100*99*98*97*96). By multiplying this probability by the total number of combinations, you can find the number of samples without broken widgets. Alternatively, by calculating the number of combinations for a set of 97 (removing the defective widgets) and subtracting this from the total combinations, you can find the number of samples that contain at least one defective widget. This approach provides a clear method for solving the problem.
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I am working on a problem in which there are 100 widgets. I need to find how many samples of size 5 there are, which I found to be 75,287,520. No problem there. I am not sure how to go about the next part, however. In the original set of 100 widgets, 3 are broken. How many of the samples of 5 contain at least one of these broken widgets? No idea how to do that. Can someone please help? Thanks!
 
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The probability that a given set of 5 widgets has no broken ones is
(97*96*95*94*93)/(100*99*98*97*96). Multiply that by the total number of combinations to get the number without any broken.
 
If you want to use the basic counting formula, remove the 3 bad widgets from the set and you have 97. Calculate the number of samples of size 5 for the 97 and subtract from the number of samples of 5 for 100.
 

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