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!
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....
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.