Suppose 36 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 the three chosen objects are adjacent nor diametrically opposite?


Total ways of 3 selections = ${C_{3}}^{36}$

Three at adjacent positions = $36$

Exactly two at consecutive positions = $36 * 32$

Diametrically opposite but not adjacent = $\frac{36 * 30}{2}$

So ${C_{3}}^{36} – 36 – 36*32 – 18*30 = 5412$


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