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.

 

[blue_raver22]

Hello,

Thank you for sharing your problem with us. Calculating the number of possible hand gestures can be a challenging task, but it is also an interesting mathematical problem.

To solve this problem, we need to break it down into smaller components. First, let's consider the number of fingers that can be raised. As you mentioned, there are three options for each hand (right, left, or both) and five fingers on each hand. This gives us a total of 3*5 = 15 options.

Next, we need to consider the possibility of raising a fist as a gesture. This would add an additional option for each hand, giving us a total of 3*6 = 18 options.

Now, let's think about the possibility of extending some fingers while keeping others down. This would give us a combination of raised and unraised fingers, and we can use Pascal's Triangle to calculate the number of possible combinations. For example, if we want to know how many ways we can raise 2 fingers on a hand with 5 fingers, we can use the formula (5 choose 2) = 5!/(2!(5-2)!) = 10.

Using this approach, we can calculate the number of possible hand gestures as:

15 + 18 + (5 choose 1) + (5 choose 2) + (5 choose 3) + (5 choose 4) + (5 choose 5) = 121

Therefore, there are a total of 121 possible hand gestures based on the given criteria. I hope this helps you in solving your problem. Best of luck!