Weird binomial coefficients

September 27, 2016

The binomial coefficients find great many uses in combinatorics, but also in calculus. The usual way we understand the binomial coefficients is

\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!},

where n and k are integers. But what do you do with \displaystyle \binom{\frac{1}{2}}{k}?! Is it even defined?

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On rectangles

February 19, 2013

While reading Cryptography: The Science of Secret Writing [1], the author makes a remark (p. 36) (without substantiating it further) that the number of possible distinct rectangles composed of n squares (a regular grid) is quite limited.

Mondrian_CompRYB

Unsubstantiated (or, more exactly, undemonstrated) claims usually bug me if the demonstration is not immediate. In this case, I determined fairly rapidly the solution, but I guess I would have liked something less vague than “increased in size does not necessarily provide greater possibilities for combination.” Let’s have a look at that problem.

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