TITLE: In how many ways can two octopi shake hands?
QUESTION [5 upvotes]: Given the question "In how many ways can two octopi shake hands?" I want to compute the different combination of handshakes. 
There are obviously many ways to interpret the question (ie. what even constitutes a handshake between 2 octopi). 
In the easiest interpretation, a handshake could be just the contact of one hand/tentacle to one other tentacle. In this case, the solution is simply:
$\binom 81 \cdot \binom 81 = 64$. 
I'm looking for a way to count the number of ways to have at least one handshake (to have 1 or more handshakes simultaneously). How would I do that?

REPLY [2 votes]: You've solved it for the optopi doing 1 handshake, the next question is how many possible handshakes are there if each octopi uses 2 tentacles.
Firstly how many ways can you choose 2 tentacles from 8? 8C2=28
So there are 28 combinations for the first octopus' tentacles. When it is shaking the second octopus' tentacle it will have 8 options when choosing the first tentacle to grab and 7 options when choosing a second tentacle, that is 8x7=56 options for the second octopus' tentacle.
That makes for a total of 28 x 56 = 1568 ways to shake 2 tentacles at once.
You can repeat this approach for shaking 3,4,5...8 tentacles at once and sum them all together