Find the number of ways of choosing three circles below, so that no two circles are next to each other.

bader Feb 14, 2024

#1**0 **

We can do this by casework:

Case 1: CORNERS

Each corner has \(5\) circles we can choose that are not next to it (The 5 on the border). There are 4 corners so, \(4\times5=20\)

Case 2: EDGES

Each edge has 3 circles we can choose that are not next to it (The 3 opposite to it). There are 4 edges so, \(4\times3=12\)

so: \(20+12=\boxed{32}\)

Note: there woudve been a third case with the center but the center is next to all the cricles in a 3x3 grid

ggwp :)

EnormousBighead Feb 15, 2024