Martial Arts Combinations

5/22/11

Consider your martial arts moves consisting of only Left Punch (LP), Right Punch (RP), Left Kick (LK), and Right Kick (RP). How many ways can you use these movements in say 1 to 4 moves?

Note, the order of moves is important (ie. LP then RP is different from RP then LP) and repetition of movements is allowed (ie. LP LP is allowed).

Done singly, you have 4 ways:

{LP} {RP} {LK} {RK}

If you string together two movements, you have 16 ways:

{LP,LP} {LP,RP} {LP,LK} {LP,RK} {RP,LP} {RP,RP} {RP,LK} {RP,RK} {LK,LP} {LK,RP} {LK,LK} {LK,RK} {RK,LP} {RK,RP} {RK,LK} {RK,RK}

What about 3 in a row? There are 64 ways:

{LP,LP,LP} {LP,LP,RP} {LP,LP,LK} {LP,LP,RK} {LP,RP,LP} {LP,RP,RP} {LP,RP,LK} {LP,RP,RK} {LP,LK,LP} {LP,LK,RP} {LP,LK,LK} {LP,LK,RK} {LP,RK,LP} {LP,RK,RP} {LP,RK,LK} {LP,RK,RK} {RP,LP,LP} {RP,LP,RP} {RP,LP,LK} {RP,LP,RK} {RP,RP,LP} {RP,RP,RP} {RP,RP,LK} {RP,RP,RK} {RP,LK,LP} {RP,LK,RP} {RP,LK,LK} {RP,LK,RK} {RP,RK,LP} {RP,RK,RP} {RP,RK,LK} {RP,RK,RK} {LK,LP,LP} {LK,LP,RP} {LK,LP,LK} {LK,LP,RK} {LK,RP,LP} {LK,RP,RP} {LK,RP,LK} {LK,RP,RK} {LK,LK,LP} {LK,LK,RP} {LK,LK,LK} {LK,LK,RK} {LK,RK,LP} {LK,RK,RP} {LK,RK,LK} {LK,RK,RK} {RK,LP,LP} {RK,LP,RP} {RK,LP,LK} {RK,LP,RK} {RK,RP,LP} {RK,RP,RP} {RK,RP,LK} {RK,RP,RK} {RK,LK,LP} {RK,LK,RP} {RK,LK,LK} {RK,LK,RK} {RK,RK,LP} {RK,RK,RP} {RK,RK,LK} {RK,RK,RK}

And lastly, doing 4 in a row yields 256 ways:

{LP,LP,LP,LP} {LP,LP,LP,RP} {LP,LP,LP,LK} {LP,LP,LP,RK} {LP,LP,RP,LP} {LP,LP,RP,RP} {LP,LP,RP,LK} {LP,LP,RP,RK} {LP,LP,LK,LP} {LP,LP,LK,RP} {LP,LP,LK,LK} {LP,LP,LK,RK} {LP,LP,RK,LP} {LP,LP,RK,RP} {LP,LP,RK,LK} {LP,LP,RK,RK} {LP,RP,LP,LP} {LP,RP,LP,RP} {LP,RP,LP,LK} {LP,RP,LP,RK} {LP,RP,RP,LP} {LP,RP,RP,RP} {LP,RP,RP,LK} {LP,RP,RP,RK} {LP,RP,LK,LP} {LP,RP,LK,RP} {LP,RP,LK,LK} {LP,RP,LK,RK} {LP,RP,RK,LP} {LP,RP,RK,RP} {LP,RP,RK,LK} {LP,RP,RK,RK} {LP,LK,LP,LP} {LP,LK,LP,RP} {LP,LK,LP,LK} {LP,LK,LP,RK} {LP,LK,RP,LP} {LP,LK,RP,RP} {LP,LK,RP,LK} {LP,LK,RP,RK} {LP,LK,LK,LP} {LP,LK,LK,RP} {LP,LK,LK,LK} {LP,LK,LK,RK} {LP,LK,RK,LP} {LP,LK,RK,RP} {LP,LK,RK,LK} {LP,LK,RK,RK} {LP,RK,LP,LP} {LP,RK,LP,RP} {LP,RK,LP,LK} {LP,RK,LP,RK} {LP,RK,RP,LP} {LP,RK,RP,RP} {LP,RK,RP,LK} {LP,RK,RP,RK} {LP,RK,LK,LP} {LP,RK,LK,RP} {LP,RK,LK,LK} {LP,RK,LK,RK} {LP,RK,RK,LP} {LP,RK,RK,RP} {LP,RK,RK,LK} {LP,RK,RK,RK} {RP,LP,LP,LP} {RP,LP,LP,RP} {RP,LP,LP,LK} {RP,LP,LP,RK} {RP,LP,RP,LP} {RP,LP,RP,RP} {RP,LP,RP,LK} {RP,LP,RP,RK} {RP,LP,LK,LP} {RP,LP,LK,RP} {RP,LP,LK,LK} {RP,LP,LK,RK} {RP,LP,RK,LP} {RP,LP,RK,RP} {RP,LP,RK,LK} {RP,LP,RK,RK} {RP,RP,LP,LP} {RP,RP,LP,RP} {RP,RP,LP,LK} {RP,RP,LP,RK} {RP,RP,RP,LP} {RP,RP,RP,RP} {RP,RP,RP,LK} {RP,RP,RP,RK} {RP,RP,LK,LP} {RP,RP,LK,RP} {RP,RP,LK,LK} {RP,RP,LK,RK} {RP,RP,RK,LP} {RP,RP,RK,RP} {RP,RP,RK,LK} {RP,RP,RK,RK} {RP,LK,LP,LP} {RP,LK,LP,RP} {RP,LK,LP,LK} {RP,LK,LP,RK} {RP,LK,RP,LP} {RP,LK,RP,RP} {RP,LK,RP,LK} {RP,LK,RP,RK} {RP,LK,LK,LP} {RP,LK,LK,RP} {RP,LK,LK,LK} {RP,LK,LK,RK} {RP,LK,RK,LP} {RP,LK,RK,RP} {RP,LK,RK,LK} {RP,LK,RK,RK} {RP,RK,LP,LP} {RP,RK,LP,RP} {RP,RK,LP,LK} {RP,RK,LP,RK} {RP,RK,RP,LP} {RP,RK,RP,RP} {RP,RK,RP,LK} {RP,RK,RP,RK} {RP,RK,LK,LP} {RP,RK,LK,RP} {RP,RK,LK,LK} {RP,RK,LK,RK} {RP,RK,RK,LP} {RP,RK,RK,RP} {RP,RK,RK,LK} {RP,RK,RK,RK} {LK,LP,LP,LP} {LK,LP,LP,RP} {LK,LP,LP,LK} {LK,LP,LP,RK} {LK,LP,RP,LP} {LK,LP,RP,RP} {LK,LP,RP,LK} {LK,LP,RP,RK} {LK,LP,LK,LP} {LK,LP,LK,RP} {LK,LP,LK,LK} {LK,LP,LK,RK} {LK,LP,RK,LP} {LK,LP,RK,RP} {LK,LP,RK,LK} {LK,LP,RK,RK} {LK,RP,LP,LP} {LK,RP,LP,RP} {LK,RP,LP,LK} {LK,RP,LP,RK} {LK,RP,RP,LP} {LK,RP,RP,RP} {LK,RP,RP,LK} {LK,RP,RP,RK} {LK,RP,LK,LP} {LK,RP,LK,RP} {LK,RP,LK,LK} {LK,RP,LK,RK} {LK,RP,RK,LP} {LK,RP,RK,RP} {LK,RP,RK,LK} {LK,RP,RK,RK} {LK,LK,LP,LP} {LK,LK,LP,RP} {LK,LK,LP,LK} {LK,LK,LP,RK} {LK,LK,RP,LP} {LK,LK,RP,RP} {LK,LK,RP,LK} {LK,LK,RP,RK} {LK,LK,LK,LP} {LK,LK,LK,RP} {LK,LK,LK,LK} {LK,LK,LK,RK} {LK,LK,RK,LP} {LK,LK,RK,RP} {LK,LK,RK,LK} {LK,LK,RK,RK} {LK,RK,LP,LP} {LK,RK,LP,RP} {LK,RK,LP,LK} {LK,RK,LP,RK} {LK,RK,RP,LP} {LK,RK,RP,RP} {LK,RK,RP,LK} {LK,RK,RP,RK} {LK,RK,LK,LP} {LK,RK,LK,RP} {LK,RK,LK,LK} {LK,RK,LK,RK} {LK,RK,RK,LP} {LK,RK,RK,RP} {LK,RK,RK,LK} {LK,RK,RK,RK} {RK,LP,LP,LP} {RK,LP,LP,RP} {RK,LP,LP,LK} {RK,LP,LP,RK} {RK,LP,RP,LP} {RK,LP,RP,RP} {RK,LP,RP,LK} {RK,LP,RP,RK} {RK,LP,LK,LP} {RK,LP,LK,RP} {RK,LP,LK,LK} {RK,LP,LK,RK} {RK,LP,RK,LP} {RK,LP,RK,RP} {RK,LP,RK,LK} {RK,LP,RK,RK} {RK,RP,LP,LP} {RK,RP,LP,RP} {RK,RP,LP,LK} {RK,RP,LP,RK} {RK,RP,RP,LP} {RK,RP,RP,RP} {RK,RP,RP,LK} {RK,RP,RP,RK} {RK,RP,LK,LP} {RK,RP,LK,RP} {RK,RP,LK,LK} {RK,RP,LK,RK} {RK,RP,RK,LP} {RK,RP,RK,RP} {RK,RP,RK,LK} {RK,RP,RK,RK} {RK,LK,LP,LP} {RK,LK,LP,RP} {RK,LK,LP,LK} {RK,LK,LP,RK} {RK,LK,RP,LP} {RK,LK,RP,RP} {RK,LK,RP,LK} {RK,LK,RP,RK} {RK,LK,LK,LP} {RK,LK,LK,RP} {RK,LK,LK,LK} {RK,LK,LK,RK} {RK,LK,RK,LP} {RK,LK,RK,RP} {RK,LK,RK,LK} {RK,LK,RK,RK} {RK,RK,LP,LP} {RK,RK,LP,RP} {RK,RK,LP,LK} {RK,RK,LP,RK} {RK,RK,RP,LP} {RK,RK,RP,RP} {RK,RK,RP,LK} {RK,RK,RP,RK} {RK,RK,LK,LP} {RK,RK,LK,RP} {RK,RK,LK,LK} {RK,RK,LK,RK} {RK,RK,RK,LP} {RK,RK,RK,RP} {RK,RK,RK,LK} {RK,RK,RK,RK}

Note that 4+16+64+256 = 340, so there are 340 possible ways to string together 1-4 moves.

To generalize, we have a geometric series that sums to S (=340 for us). The first element is a (4 for us), the factor between elements is r (4 for us. For example 4*4 = 16, and 16*4 = 64, etc.), and n is the length of the list (4 for us).

S=a+ar+ar²+ar³+...+ar^n-1

Sr=ar+ar²+ar³+...+ar^n-1+arⁿ

S-Sr=a-arⁿ [subtracting the second line from the first]

S(1-r)=a(1-rⁿ)

S=a(1-rⁿ)/(1-r)

Substituting in, we get S = 4(1-4⁴)/(1-4) = 340

To generalize, if you know n moves, the total number of possible combinations of stringing together these moves 1 to n in a row is given by

S=sum(n^i, i=1 to n) = (nⁿ⁺¹-n) / (n-1)

• n = 1, S = 1 (by definition)
• n = 2, S = 6
• n = 3, S = 39
• n = 4, S = 340
• n = 5, S = 3,905
• n = 6, S = 55,986
• n = 7, S = 960,799
• n = 8, S = 19,173,960
• n = 9, S = 435,848,049
• n = 10, S = 11,111,111,110