there are 30 people at a party. each person shakes hands with every other person.... how many handshakes take place.
is it something like the 1st person shakes hands with the 3, 5, 7, 9, 11, 13, 15, 17, 19 21, 23, 25, 27 29
2nd person 4, 6, 8, 10, 12, 14, 16, 18, 20 22, 24, 26, 28 30?
"each person shakes hands with every other person...." does not literally mean shaking the hand of alternate people such as you indicate by 3, 5, 7, 9, ...etc. It meant that you shake the hand of each of the other people at the party.
Lets look at a typical problem with other numbers.
If there are 15 people in a room and each person shakes hands with every other person in the room only once, how many handshakes will take place?
There are two ways of looking at this type of problem. The first involves determining the number of combinations of "n" things taken "r" at a time and the other the sum of all possible pairings of "n" things.
For the first method, what you are seeking is the number of possible combinations of 15 people taken two at a time with no combination being repeated in another order. What I mean here is that once "a" shakes the hand of "b", then that pairing has been made and cannot be repeated in the form of "b" shaking the hand of "a" at another time. Only the single pairings are counted. This then boils down to determining the number of possible combinations of "n" different things taken "r" at a time, C(n,r), which is derivable from the formula C(n,r) = [n(n-1)(n-2)-----(n-r+1)/r! = n!/[r!(n-r)!], where n! = n(n-1)(n-2)------(3)(2)(1), r! means r(r-1)(r-2) etc., and similarly for (n-r)!. So in your problem n = I15 and r = 2. Thus you have C(15,2) = [15x14x13----3x2x1]/(2x1)(13x12x11----3x2x1) = 15x14/2x1 = 210/2 = 105 handshakes.
The second method seeks the sum of all possible pairings between the n people as follows, using 5 people to illustrate the derivation. Person 1 shakes the hand of 4 other people. Person 2 shakes the hands of 3 other people already having shaken the hand of person 1. Person 3 shakes the hands of 2 other people already having shaken the hands of persons 1 and 2. Person 4 shakes the hand of person 5, also having already shaken the hands of persons 1, 2, and 3. So what do we have in the way of a total? All together, 4 + 3 + 2 +1 = 10.
What we did in fact is simply sum up the numbers from 4 to 1, or (n - 1) to 1. Now, the sum of any sequence of consecutive numbers from one on up is given by S = n(n + 1)/2. Notice that our sum starts with (n - 1). Therefore, the total number of handshakes between "n" people can be defined as S = [(n - 1)(n - 1 + 1)]/2 or (n(n - 1)/2.
Applying this to our 15 person problem, S = 15(14)/2 = 210/2 = 105 handshakes.
