I need help with the following question which I cannot seem to solve:
17 students are sitting in a circle. Each person shakes hands with everyone but his/her neighbours. How many handshakes have been exchanged?
My approach: no. of ways $ = 1 + 2 + 3 + … + 14 = 7(15) = 105 $.
Apparently my answer is wrong (correct ans is 119). But I can't seem to understand why. Could someone please explain?
Best Answer
Each of 17 people shakes hands with 14 people (all except themselves and their 2 neighbors), so there are
$$\frac{17\times 14}{2} = 119$$
handshakes (dividing by 2 to account for symmetry, as you would otherwise count "$A$ shaking hands with $B$" and "$B$ shaking hands with $A$" as distinct events).