How many ways can a drummer with a varying amount of limbs play?

Question

I’m looking at the amount of possible combinations of single stroke rolls using varying amount of limbs.

Let’s use these letters for reference

• R is right hand.
• L is left hand.
• r is right foot.
• l is left foot.

There is one variation for one limb: R. There are two variations for two limbs: RL
and LR. There are six variations for three limbs:

• RLr
• RrL
• LRr
• LrR
• rRL
• rLR

I won’t list them all, but there are 24 combinations for four limbs. Here is the first one:

• RLrl
• RLlr
• RrLl
• RrlR
• RlLr
• RlrL

That leads to the following ratios: $1:1$, $2:2$, $3:6$ and $4:24$. That would leave me to believe the next pair of numbers to continue the pattern would be $5:120$.

Is there an equation that I can use to solve for number of variations using different numbers of limbs using only one limb once?

Answer 1

You are trying to count, for each fixed number $n$ the number of ways to arrange the numbers $1, 2, 3, \dots, n$ in a sequence, using each number exactly once. These are called permutations of $1, 2, 3, \dots, n$. The number of permutations of $1, 2, 3, \dots, n$ is called "$n$ factorial", written $n!$, and is equal to $n! = (n) (n-1)(n-2) \cdots (2) (1)$. The reason is that you have $n$ choices to fill the first position, $n-1$ choices to fill the second position, etc.

Answer 2

The pattern you posted so far matches:

F(1) = 1
F(X) = X * F(X-1)

Answer 3

The problem with all of the other answers you've received so far, is that you only have four limbs. Based on the rules you have implied in your question, I see no reason to assume that there is only one way for the first variation. You could, for example, do "R", but you could also do "L", or "l", or "r"--so in this case, I count four possibilities, not one.

Similarly, for two limbs, you seem to ignore the feet, but give no reason why. For three limbs, you ignore the left foot.

In the case of using all four limbs, with the exception of the fourth outcome (which seems to be a typo), all of the outcomes you enumerated use "R", "L", "r", "l" exactly once, but it seems that order is important--at least, it would be if you count 24 outcomes as you claim.

Finally, you ask:

Is there an equation that I can use to solve for number of variations using different numbers of limbs using only one limb once? (emphasis mine)

This is clearly problematic: if you can use each limb only once, it is impossible to have a variation that uses more than four limbs, unless you are not human, or have extra limbs, or are counting some other appendage as a limb. So you need to think about what you actually mean by this statement.

For what it's worth, I count 4 variations for a single hit on the drum, 12 variations for two hits, 24 variations for three hits, and 24 variations for using all four limbs. Variations with five or more hits are not possible--you don't have five limbs to use.