# Suppose 32 objects are placed along a circle at equal distance.

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Suppose 32 objects are placed along a circle at equal distance. In how many ways can 3 objects be chosen from among them so that no two of three chosen objects are adjacent nor diametrically opposite?

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One can chose 3 objects out of 32 objects in (323) ways. Among these choices all would be together in 32 cases; exactly two will be together in 32 x 28 cases. Thus three object can be chosen such that no two adjacent in (323) - 32 - (32 x 28) ways. Among these, further, two objects will be diametrically opposite in 16 ways and third would be on either semicircle in a non adjacent portion in 32 – 6 = 26 ways. Thus required number is

(323) - 32 - (32 x 28) - (16 x 26) = 3616