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# Counting

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This is due really soon, I need it

At a meeting, five scientists, two mathematicians, five journalists, two biologists, and three politicians are to be seated around a circular table.  How many different arrangements are possible if the scientists must all sit together (in five consecutive seats).  (Two seatings are considered equivalent if one seating can be obtained from rotating the other.)

Mar 6, 2023

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We can treat the block of 5 scientists as a single entity and arrange them in a line first. This can be done in 5! ways. Then, we can arrange the other 12 people around the circular table in (12-2+1)! = 11! ways (since we need to leave 2 seats for the mathematicians who will sit together).

Finally, since the table is circular, we need to divide by 14 (the number of positions we can rotate the table to get the same arrangement) to get the total number of distinct arrangements:

5! * 11! / 14

Simplifying, we get:

(5 * 4 * 3 * 2 * 1) * (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / 14

= 120 * 39,916,800 / 14

= 342,720,000

Therefore, there are 342,720,000 different arrangements possible if the scientists must all sit together.

Mar 6, 2023