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This may look like a neatly arranged stack of numbers, but it’s actually a mathematical treasure trove. It is known as the Pascal’s Triangle, after French mathematician Blaise Pascal. So what is it about this that has so intrigued mathematicians the world over? In short, it’s full of patterns and secrets. We will discuss some of the secrets here. 1. Generation of Pascal Triangle: First and foremost, there’s the pattern that generates it. Start with 1 and imagine invisible zeros on either side of it. Add them together in pairs and you will generate next row. Now do that again and again, keep going and you will wind up with something like this. Though really Pascal’s triangle goes on infinitely. 2. Binomial Expansion of the form x y: Now each row corresponds to what’s called the coefficients of a Binomial expansion of the form x plus y raised to the n (x y)n, where n is the number of the row and we start counting from zero. So if you make n equal to 2 (n=2)and expand it, you get x² 2xy y². The coefficient or numbers in front of the variables are the same as the numbers in that row of Pascal’s triangles. You will see the same thing with n = 3which expands to this. So the triangle is a quick and easy way to look up all of these coefficients, but there’s much more. of 2 : Now let’s take a look at powers of 2. If you notice, the sum of the numbers in Row 0 is 1 or 2^0. Similarly, in Row 1, the sum of the numbers is 1 1 = 2 = 2^1. If you will look at each row down to row 5, you will see that this is true. In fact, if Pascal’s triangle was expanded further past Row 5, you would see that the sum of the numbers of any nth row would equal to 2^n. #PascalsTriangle #math #probability #Genetics
