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

 

 Feb 14, 2024
 #1
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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 :)

 Feb 15, 2024
edited by EnormousBighead  Feb 15, 2024

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