Math, math, math, and more math!: Symmetry, Reflection, and When to Divide

Saturday, October 28, 2017

Symmetry, Reflection, and When to Divide

In topic: Counting with Symmetry (Counting & Probability).

Incorrect. Oops!

In how many ways can $7$ people sit around a round table if no two of the $3$ people Pierre, Rosa, and Thomas can sit next to each other?

Your First Answer: 384

Your Second Answer: 1008

Solution:

After Pierre sits, we can place Rosa either two seats from Pierre (that is, with one seat between them) or three seats from Pierre. We tackle these two cases separately:

Case 1: Rosa is two seats from Pierre. There are $2$ such seats. For either of these, there are then four empty seats in a row, and one empty seat between Rosa and Pierre. Thomas can sit in either of the middle two of the four empty seats in a row. So, there are $2\cdot 2 = 4$ ways to seat Rosa and Thomas in this case. There are then $4$ seats left, which the others can take in $4! = 24$ ways. So, there are $4\cdot 24 = 96$ seatings in this case.

Case 2: Rosa is three seats from Pierre (that is, there are $2$ seats between them). There are $2$ such seats. Thomas can't sit in either of the $2$ seats directly between them, but after Rosa sits, there are $3$ empty seats in a row still, and Thomas can only sit in the middle seat of these three. Once again, there are $4$ empty seats remaining, and the $4$ remaining people can sit in them in $4! = 24$ ways. So, we have $2\cdot 24 = 48$ seatings in this case.

Putting our two cases together gives a total of $96+48 = \boxed{144}$ seatings.