Family Dinner
The Jones own only one table, and it's perfectly circular. Mr. Jones is setting the table for dinner tonight and he was wondering, "how many ways can my family sit down at the table?" Mr. Jones has five children, a wife, and a grandmother all coming to dinner, and rotations of the seating arrangement are not considered to be different.
The answer is 5040.
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Let's assign each family member a number 1 through 8. Then this problem sort of sounds like counting permutations ( n ! ), but there's this weird fact that rotations aren't different from each other. So 1,2,3,4 is the same as 2,3,4,1. Lets let the number we're after be x, and figure a way to get to counting permutations using x.
Notice that we can get all permutations of [8] by unwrapping the circle of seats starting at each position on the table. For each particular seating arrangement, we could unwrap the seats starting at 8 different places and so 8 x = 8 ! ⟹ x = 7 ! = 5 0 4 0 .