Discrete math problem college level question

[afang]

Suppose 2n people sit on a round table and are shaking hands in
pairs. Suppose that etiquette is observed and no 2 shakes cross. Let
S_n be the number of possible shaking hands arrangements of this sort.

Determine S_10.

 

[phoenixthoth]

I recommend trying to work this out for n=2, 3, and 4 and then find a pattern.

You'll need to find a formula for the sum of the first k whole numbers which is probably in your book somewhere.

 

[M98Ranger]

To solve this problem, we can use the concept of permutations and combinations. Since each person can only shake hands with one other person at a time, the total number of handshakes that can occur is n (the number of people) multiplied by (n-1) (the number of potential handshakes for each person). This can be written as n(n-1).

However, this calculation includes both clockwise and counterclockwise handshakes, which are considered the same in this scenario. Therefore, we need to divide the total number of handshakes by 2 to account for this repetition. This gives us the formula:

S_n = n(n-1)/2

Substituting n=10 into this formula, we get:

S_10 = 10(10-1)/2 = 45

Therefore, there are 45 possible ways for 10 people to shake hands in pairs around a round table while observing etiquette and without any crossed handshakes.