### Question 1

Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3objects be chosen from among them so that no two

of the three chosen objects areadjacent nor diametrically opposite ?

Answer: 3616

### Comments on Question 1

### Solution

Soln1 : First select any one point and then from remaining 29 points (except that point andtwo neighbouring points) select two points such

that they are not consecutive then numbersof ways will be: (^{32}C _{1} x ^{28}C _{2})/3

Now subtract the number of ways of selection of diametrically opposite points.So total number of ways

= ((^{32}C _{1} x ^{28}C _{2})/3 )-(^{16}C _{1} x ^{26}C _{1})

= 4032-416=3616