Combinatorics: MasterChef

Six equally-talented chefs, labeled 1 to 6, are competing in a cooking contest. What is the probability that Chef 2 finishes in second place and Chef 5 ranks among the top three?

Answer

There are 6! ways to order them in total (total outcomes). Now, for the favorable outcomes:

\[ \text{Case 1: place 1 and 2 fixed... the rest can be ordered in }4! \]
\[ \text{Case 2: place 2 and 3 fixed... the rest can be ordered in }4! \]

Hence,

\[ \frac{2 \times 4!}{6!} = \frac{1}{15}. \]
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