Non-crossing handshakes for 20 people at round table

[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.

 

[ahrkron]

Welcome to the forums!

Try with small numbers first, adding one person at a time. How many more handshakes are needed per added person? After the first few, you should be able to see a pattern.

 

[afang]

I do not know how to procede. The question is unclear to me. For S_4 is the answer 4 ways? Thanks.

 

[Gokul43201]

Draw a circle with 2n numbered points on its circumference...say with n=2,4,6,... etc. Now draw lines connecting these points such that (i) all points are connected - this is not stated, but I'm ammuming it is implicit, and (ii) no two lines intersect. After having drawn the lines, list the end-points of these lines as a set of unordered pairs.

How many such different sets are there, for a given n, and a given ordering of the people (don't change their positions, only change the lines) ?

Do you see a pattern across different values of n ?

(If my interpretation of this problem is correct, S(1) = 1, S(2) = 2, ...you find the rest)

 

[afang]

I guess with that patter then the number of possible shaking hand arrangements = n.
I guess I need to ask my professor to clarify the question if all people are shaking.

 

[afang]

Hello does anyone have a faster way using probability to solve this problem? Counting it out would be difficult. I know however, that the pattern has been
S_1 = 1
S_2 = 2
S_3 = 4
etc...