Question: A circular table has 7 scientists (3 biologists, 2 physicists, 2 chemists). How many seating arrangements are possible if scientists of the same discipline must sit together?

February 22, 2026

Question: A circular table has 7 scientists (3 biologists, 2 physicists, 2 chemists). How many seating arrangements are possible if scientists of the same discipline must sit together?

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Solution: First, compute the total number of ways to choose 3 species: $\binom{20}{3} = 1140$. The number of favorable outcomes is $\binom{10}{2} \cdot \binom{6}{1} = 45 \cdot 6 = 270$. The probability is:
\frac{270}{1140} = \frac{9}{38}
The probability is $\boxed{\dfrac{9}{38}}$.
Solution: Treat each discipline as a block. There are $3!$ ways to arrange the blocks around the table. Within the biology block: $3!$, physics: $2!$, and chemistry: $2!$. For circular permutations, fix one block to eliminate rotational symmetry:
The number of valid seating arrangements is $\boxed{48}$.
Question: An entomologist identifies 8 ant species and 5 beetle species. If she samples 4 species randomly, what is the probability that exactly 2 are ants and 2 are beetles?

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