Counting Number of Possible Hand Gestures

[Armbru35]

Hi All,

I'm terribly stuck on this problem. We were asked to calculate how many hand gestures are possible, keeping in mind that a hand gesture consists of raising one or both hands and extending some fingers (note: raising just a fist is also considered a gesture). I started this problem by thinking of doing 3*5 (representing 1 finger raised on right, left or both hands) * 3*((5|10)) but something seems off. My professor suggested looking at Pascal's Triangle, but I'm not sure where to go from there. Any suggested would be so helpful! Thanks!

 

[Deveno]

let's just look at the possible gestures involving a single hand.

each gesture might involve 0 - 5 fingers.

for 0 fingers, there is one possible gesture ("the fist").

for 1 finger, there are 5 possible gestures (assuming we count the "thumb" as a finger)

in general, for k fingers there are 5 choose k (i will write this as 5Ck).

the general formula for this is 5!/(k!(5-k)!).

for 5C3, this is 120/(6*2) = 10, for example.

so we have: 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 gestures

that is, 1 + 5 + 10 + 10 + 5 + 1 = 32 (this is the sum of the 5th row of pascal's triangle).

i think you can take it from here...

 

[Cant or Wont]

I like to think I'm capable of expressing myself using my hands in an infinite number of ways.

 

[Amaranth]

How many gestures with the restriction that you cannot extend a ring finger unless you also extend the middle finger next to it?

How many gestures with the restriction that you may not extend the middle finger alone on either hand?


I can get the numbers for 0,1,2,8,9,10 fingers for each by writing out combinations but I can't come up with a simpler mathematical solution for anything in the middle. It just seems like there's too many cases to consider for each. But perhaps I'm overthinking it.