The problem I am working is the following; please provide step-by-step to obtain solution's.

I am unable to figure (c) out the ANSWER IS ... 7805

There are 5 rotten plums in a crate of 25 plums, How many samples of 4 of the 25 plums contain at least one rotten plum?

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However you seem to be on the right track (looking at your correct answers to a and b), so try to solve it this way.
the number of possibilities to pick AT ...

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The solution explains how to calculate the probabilities of several events.

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