Reuben has a 4x4 checkerboard. He uses scissors to cut out each square from the board. He then randomly arranges the 16 pieces into four rows and four columns.
Disregarding rotations, determine the probability that this layout is in a precise checkerboard pattern.
How will the answer change if rotations are taken into consideration?
I got the same result, but I thought about it a little differently: I only worried about the black pieces of paper.
There are 8! ways to correctly place the eight black squares of paper onto the correct 8 spots, while there are 16 * 15 * ... * 9 = 16!/8! ways to place these 8 squares anywhere onto the 16 spots. The ratio is 8!^2/16! = 0.000777, the answer. Doubling this number is the probability of landing all 8 black paper squares either into the black 8 or white 8 spots (i.e, a rotation).
In the notation of combinatorics, this would be phrased:
p = (8 permute 8) / (16 permute 8) = 8P8 / 8P16 =
1 / 16C8 = 1/ (16 choose 8) = 1 / (16 8) = 1 / [16! / ((16-8)! 8!) ]
Edited on May 18, 2022, 1:07 am
soln
