['Y', 'L', 'L', 'A', 'C', 'I', 'H', 'P'],

['L', 'L', 'A', 'C', 'I', 'H', 'P', 'A'],

['L', 'A', 'C', 'I', 'H', 'P', 'A', 'R'],

['A', 'C', 'I', 'H', 'P', 'A', 'R', 'G'],

['C', 'I', 'H', 'P', 'A', 'R', 'G', 'O'],

['I', 'H', 'P', 'A', 'R', 'G', 'O', 'I'],

['H', 'P', 'A', 'R', 'G', 'O', 'I', 'L'],

['P', 'A', 'R', 'G', 'O', 'I', 'L', 'B'],

['A', 'R', 'G', 'O', 'I', 'L', 'B', 'I'],

['R', 'G', 'O', 'I', 'L', 'B', 'I', 'Z']

"ZIBLIOGRAPHICALLY" is the word that I want to know how many times it can be found in this square. Starting from the bottom right corner, and working your way up to the top left corner. How would I go about figuring this out?


It looks like you want to count North-West paths on a lattice.

You'll have to make $9$ steps to the north, and $7$ steps to the west, and you can do so in any order. This means that there are $\displaystyle \binom{9 + 7}{7} = \frac{16!}{16!7!} = 11440$ ways of doing this.

  • 1
    $\begingroup$ Careful with the off-by-one errors. In the situation at hand, there are only $10-1$ northward steps, and only $8-1$ westward steps, so $$\frac{((10-1)+(8-1))!}{(10-1)!\,(8-1)!}=\frac{16!}{9!7!}=11440$$ paths. Note that this settles the issue in @Jacob's (now-deleted) comment, since $$\frac{((2-1)+(2-1))!}{(2-1)!\,(2-1)!}=\frac{2!}{1!1!}=2$$ $\endgroup$
    – Blue
    Oct 24 '18 at 23:48

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