In Kung Fu, there are 32 basic movement called \"keys\" eight of each of punches

Question

In Kung Fu, there are 32 basic movement called "keys" eight of each of punches, blocks, kicks and stances. A technique is a combination of 2 or more keys. For Example, a technique could be simultaneous vertical punch with one hand and a hooking block with another from a horse stance.

(a) How many different techniques are there in which all keys are simultaneous ( provided one
can use both hands independently at the same time, but do only one kick or one stance?

(b) A step is a more complicated notion than a technique. A step always ends in a stance, preceded either by another stance or a kick. During one step each hand can do up to two keys
(punches or blocks). it does not matter whether a key is executed by left or right hand or foot.
How many different steps are possible?

(c) Now suppose that each key can be done in 8 directions, where each arrangement of directions is considered a different technique; How many steps are there in this case?

Explanation / Answer

(a) There are (8C2 + 8) = 36 ways to use your hands, and 16 ways to pick either a kick or a stance. Hence, there are in total 36*16 = 576 techniques (b) In each step, there are 37 ways to use your hands (the 36 from the last part, or doing nothing with your hands). The last part of the technique is one of 8 stances, and the first part is one of 16 stances or kicks. Therefore, there are in total 37*8*16 = 4736 techniques (c) In the first part of the step you can do one of 16 stance or kicks, in one of 8 directions. On the second part of the step, you can do one of 8 stances, in one of 8 directions. This makes a total of 16*8*8*8 = 8192. But then you can additionally use your hands to go do (8C2 * 8C2 + 8*8 + 1) = 849. So in total, there are 8192*849 = 6955008 techniques

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