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  Well stated, I couldn't agree more!

  Posted by sodium11 on May 20, 2008 at 09:32:12 Pacific Time

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  sodium11, I also think the explicitly combinatorial solution is more natural. However, both approaches are two sides of the same coin since the binomial coefficients are defined by the recursion B(n,m) = B(n-1,m) + B(n-1, m-1).
  

  Posted by eharley on May 20, 2008 at 06:19:35 Pacific Time

• Cheesecake - brute force or formula?
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  While I would never disparage the brute force method of solving the cheesecake-checkerboard problem, I hardly think it is the "easiest" way to solve the problem, especially if you have a calculator. The way I generated the formula is as follows:
  
  The total number of movements the critter must make is sixteen: eight eastward and eight southward.
  
  Suppose slips of paper numbered 1 through 16 are placed in a hat and we pick out eight of them at random; and at the steps represented by those numbers, he moves east, and at the other steps he moves south. (Thus, if one of the slips is #1, then the first step will be east, if slip #1 is not picked, the first step will be south.)
  
  The total number of possible paths to the cheesecake is equal to the number of combinations of eight slips that can be picked from a set of sixteen. This is equal to 16!/8!8!, which yields the correct answer.
  
  The more general version of this formula is (l+w)!/l!w! where l is the length of the grid and w is the width.
  

  Posted by sodium11 on May 19, 2008 at 14:53:26 Pacific Time