Circular Permutations Questions

We’ve all seen those tricky Permutation/Combination questions involving people around a circular table. How do we solve them? Well, they’re actually pretty ridiculously easy!

Let’s examine one:

There are 7 people and a round table with 5 seats. How many arrangements are possible?

This question might seem complex at first because there are more people than there are seats. It’s kind of a Permutation AND a Combination in one!

So first we’re wondering, how many ways to choose 5 from 7? This is a simple Comb:

7C5 = 7! / 5!2! = 7 x 6 / 2 = 42/2 = 21 ways

So now for each of those ways, we’re wondering, how many ways can we order 5 people around a table?

For any table with “x” seats, the number of possible arrangements is (x-1)!, so here 4! = 4 x 3 x 2 = 24.

21 x 24 = 504

The correct answer is 504.

The key takeaway here is that “choosing” X from Y always allows for the Combination formula (x! / (x-y)! y!), and the number of arrangements around a circular table with “X” seats is always (X-1)! There’s actually not that much to memorize!

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