Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from amongst them so that no two of the three chosen objects are adjacent nor diametrically opposite.

One can choose 3 objects out of 36 objects in (36,3) ways. Among these choices all would be together in 36 cases; exactly two will be together in 36 x 32 cases. Thus three objects can be chosen such that no two adjacent in (36,3) - 36 - (36 x 32) ways. Among these, furthrer, two objects will be diametrically opposite in 18 ways and the third would be on either semicircle in a non adjacent portion in 36 - 6 = 30 ways. Thus required number is

 

36c3 - 36 - (36*32) - (18*30)