RMO 2015 question number 2
Probability
Level
3
suppose 32 objects are placed along a circle at equal distances in how many ways can 3 objects be chosen from among them so that no two of three chosen objects are diametrically opposite
The answer is 3616.
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1 solution
Suppose 32objects are placed along a circle at equal distances. In how many ways can 33 objects be chosen from among them so no two of the three chosen objects are adjacent nor diametrically opposite. This is again a problem in a math contest in India and this is how I tried it:
Number of ways of choosing 3 objects from 32 objects=(32 c 3)=(32 c 3) Number of ways to choose 3 points such that they are adjacent=32=32 Number of ways to choose 33 points such that two of them are adjacent=(32×2×30)=(32×2×30).......
for each point there are two ways to choose an adjacent point. For each choice there are 3030 options to choose the third point.
Number of ways to choose 33 points such that two of them are diametrically opposite=(32×30)=(32×30) Number of ways to choose 33 points from 32 equidistant points on a circle with the restrictions placed by the problem =(32 c 3)−(32+(32×2×30)+(32×30)=(32 c 3)−(32+(32×2×30)+(32×30) Am I correct in my approach?