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# 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+...+arn-1

Sr=ar+ar2+ar3+...+arn-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

Standard martial arts disclaimer: The material contained in this article is provided for informational purposes only. Before undertaking the study of any martial art, you should consult with your physician. You should be aware if you engage in martial arts you are doing so entirely at your own risk, including any present and/or future physical or psychological pain or injury that you may incur. The author of this content cannot assume any responsibility or liability for any injuries or losses that you may incur as a result of acting upon any information provided by this content, or any links to other martial arts information found herein.

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