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Multiplication in Base 10
The Multiplication Table
Pascal's triangle is so fundamental that I don't even know what to say about it … it's like trying to write about how to add. But, just in case you've been raised by wolves, here are three important ways of looking at it.
First, each element is the sum of the ones diagonally above it.
Second, the elements are binomial coefficients—the mth row of the triangle gives the coefficients in the expansion of (x+y)m. (“Binomial” means “having two parts”; here the parts we're talking about are x and y.)
Third, the elements are combinatorial numbers. The nth element on the mth row tells you the number of ways of choosing n elements from a set of m distinct elements … or, more symmetrically, the number of ways of partitioning a set of m distinct elements into subsets of size n and m-n.
By the way, if you're actually counting rows and columns, you should start from zero, not from one.
I stopped the triangle after the eighth row mainly because that's all that fits. But, coincidentally, that's also all I really know from memory; even the seventh and eighth rows I have to think about a bit. It's easy and fun to compute more rows, but I will let you do that for yourself.
One other thing that's worth pointing out but not explaining in detail: there are higher-dimensional analogues of the triangle. If you want to think about those, the first thing to do is set up a coordinate system with the origin at the top of the triangle and the axes along the two sides. Then you can add as many more axes as you like.
Multiplication in Base 7
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@ April (2004)