# Count number of paths through a matrix

Please note: this is not homework.

This is related to a previous question I have asked, Calculate number of sequences in frequency matrix.

(also the image presented on this StackOverflow question: Plot weighted frequency matrix)

The plot is from Boring, E. G. (1941). Statistical frequencies as dynamic equilibria. Psychological Review, 48(4), 279.

For the full matrix there are $2^{36}$ paths through the full matrix. Boring states:

Frequency matrix converging upon one-sixth. The diagram is for inter- dependent events with a limiting frequency of one-sixth. There are only 1,947,792 sequences in this matrix

My question: Since I am reproducing the plot but for 25 trials (instead of 36) how do I calculate the relevant figure for 25 trials rather than for 36 (which is what Boring gives)

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Thanks for putting up the correct plot! :) – Frank_Zafka Sep 8 '11 at 12:09

You take $36$ steps, and $6$ of these are to the left and $30$ are to the right. There are $\binom{36}6=1947792$ ways to choose the $6$ steps to the left. If you do the same thing for $25$ and $6$, there are $\binom{25}6=177100$ ways to choose.