Question

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 the three chosen objects are adjacent nor diametrically opposite?

Answer 1

One can choose 3 objects out of 32 objects in (32,3) ways. Among these choices all would be together in 32 cases; exactly two will be together in 32 x 28 cases. 

Thus three objects can be chosen such that no two adjacent in (32,3) - 32 - (32 x 28) ways. Among these, furthrer, two objects will be diametrically opposite in 16 ways and the third would be on either semicircle in a non adjacent portion in 32 - 6 = 26 ways. 

Thus required number is