Counting Candies

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Six candies arranged in a row

Six candies are given to 4 people at random. What is the probability that each person receives at least one candy?

This question appeared in one of our fun math problem forums. It is a classic counting and probability problem. The question was ambiguous as to whether the six candies are distinguishable or not. This got many people confused—they were divided into two sides, each side insisting the other side’s solution was wrong. To spell out the two scenarios explicitly: (1) Suppose all the candies are identical, which means we only care about how many candies each person got, what is the probability that each person received at least one candy? (2) Suppose all the candies are distinguishable, which means if one particular candy is given to Adam, it is different from the case that the same candy is given to Bob. In this situation, what is the probability that each person received at least one candy?

Do these two scenarios give the same answer after all? Please share your solutions.

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