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jmc

counting and probability senior

Problem

A Senate committee has 5 Democrats and 5 Republicans. In how many distinguishable ways can they sit around a circular table if all the members of each party all sit next to each other? (If the table is rotated, it should be counted as the same seating.)
Solution
Choose any 5 consecutive seats in which to place the Democrats -- it doesn't matter which 5 consecutive seats that we choose, since we can rotate the table. Then there are ways to place the Democrats in their seats, and ways to place the Republicans in their seats, for a total of arrangements.
Final answer
14,\!400