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# Particle man

I found the following problem at http://www.artofproblemsolving.com/Store/products/intro-counting/posttest.pdf and I think it's great:

Particle Man is at the origin in three-dimensional space. In how many ways can Particle Man take a series of 12 unit-length steps, each step parallel to one of the coordinate axes, from the origin to (3, 4, 5) without passing through the point (2, 3, 2)?

Slightly harder: what if Particle Man also is not allowed to pass through the point (1,1,1)?

Note by Tim Vermeulen
4 years ago

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At first consider the total number of ways Particle Man can take a series of 12 unit-length steps from the origin to $$(3, 4, 5)$$, i.e remove the constraint first. Consider a permutation of $$3$$ identical $$x$$s, $$4$$ identical $$y$$s, and $$5$$ identical $$z$$s. For example, $$xxyyyyxzzzzz$$ is such a permutation. In each such permutation, a $$x$$ corresponds to moving one step along $$x$$ axis, a $$y$$ corresponds to moving one step along $$y$$ axis, and a $$z$$ corresponds to moving one step along $$z$$ axis. We are talking about moving positively along the axes here, because if a negative step is taken then the total number of steps taken to reach the destination will be more than $$12$$. The total number of permutations is $$\frac{12!}{3!4!5!}$$. Now we find the number of ways Particle Man can reach his destination passing through $$(2, 3, 2)$$. Consider the first part of his journey, i.e his journey from origin to $$(2, 3, 2)$$. In a similar argument, we can prove that Particle Man can go there in $$\frac{7!}{(2!)^23!}$$ ways. Now consider the second part of his journey, i.e his journey from $$(2, 3, 2)$$ to the destination. Similarly we can prove that this can be done in $$\frac{ ((3-2) + (4-3) + (5-2))!}{(3-2)! (4-3)! (5-2)!} = \frac{5!}{3!}$$ ways. Hence the total number of ways Particle Man can reach his destination via $$(2, 3, 2)$$ is $$\frac{7!5!}{(2!)^2(3!)^2}$$ (the numbers are multiplied). We have to subtract this from the total number of ways, so the final answer will be $$\frac{12!}{3!4!5!} - \frac{7!5!}{(2!)^2(3!)^2}$$.

I will post my answer to the harder part shortly. · 4 years ago