An Explanation of Pascal's TriangleDate: 27 Feb 1995 22:57:04 -0500 From: Anonymous Subject: Probability and Pascal's Triangle My 10th grade daughter is having trouble finding a simple explanation of Pascal's Triangle and its application. So far we've discovered that, in lieu of prozac, Pascal turned to math and came up with the arithmetical triangle as an aid for his gambling buddies. But that's where our information runs out. Any help would be gratefully appreciated. Erin and Dale in Helena, Montana Date: 28 Feb 1995 14:19:21 -0500 From: Dr. Ken Subject: Re: Probability and Pascal's Triangle Hello there! Well, since I'm not quite sure how much you know about Pascal's Triangle (for instance, you seem to know that it is connected to the study of probability), I'll start pretty much from the beginning. The way I see it, Pascal's Triangle is kind of a collection of neat things in mathematics. The way you construct it follows: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 . . . You start out with the top two rows: 1, and 1 1. Then to construct each entry in the next row, you look at the two entries above it (i.e. the one above it and to the right, and the one above it and to the left). At the beginning and the end of each row, when there's only one number above, put a 1. You might even think of this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add up the number above and to the left (since there is no number there, pretend it's zero) and the number above and to the right (1), and get a sum of 1. When people talk about an entry in Pascal's Triangle, they usually give a row number and a place in that row, beginning with row zero and place zero. For instance, the number 20 appears in row 6, place 3. That's how you construct Pascal's Triangle. An interactive version where you can specify the number of rows you want to see can be found at http://mathforum.org/dr.cgi/pascal.cgi But Pascal's Triangle is more than just a big triangle of numbers. There are two huge areas where Pascal's Triangle rears its head, in Algebra and in Probability/Combinatorics. First let's look at the Algebra version. Let's say you have the polynomial 1+x, and you want to raise it to some powers, like 1,2,3,4,5,.... If you make a chart of what you get when you do these power-raisings, you'll get something like this: (x+1)^0 = 1 (x+1)^1 = 1 + x (x+1)^2 = 1 + 2x + x^2 (x+1)^3 = 1 + 3x + 3x^2 + x^3 (x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 (x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 ..... If you just look at the coefficients of the polynomials that you get, voila! Pascal's Triangle! Because of this connection, the entries in Pascal's Triangle are called the _binomial_coefficients_. They are usually written in parentheses, with one number on top of the other, for instance 20 = (6) <--- note: that should be one big set of (3) parentheses, not two small ones. I don't think it's standard notation, but when I write binomial coefficients in a text document like this, I usually write them [6:3]. The other main area where Pascal's Triangle shows up is in Probability. Let's say you have five hats on a rack, and you want to know how many different ways you can pick two of them to wear. It doesn't matter to you which hat is on top, it just matters which two hats you pick. So this problem amounts to the question "how many different ways can you choose two objects from a set of five objects?" The answer? It's the number in the second place in the fifth row, i.e. 10. (Remember that the first number is in place zero.) Because of this choosing property, the binomial coefficient [6:3] is usually read "six choose three." If you want to find out the probability of choosing one particular combination of two hats, that probability is 1/10. There's a pretty simple formula for figuring out the binomial coefficients. It's n! [n:k] = -------- k! (n-k)! 6 * 5 * 4 * 3 * 2 * 1 For example, [6:3] = ------------------------ = 20. 3 * 2 * 1 * 3 * 2 * 1 That's a basic introduction to Pascal's Triangle. It certainly also shows up in lots of other places (for example, the triangular numbers are in there, if you know what those are), but I think it would be too much for me to go into those right now. Thanks for the question! -Ken "Dr." Math |
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