0:00:07.603,0:00:11.000 This may look like a neatly arranged[br]stack of numbers, 0:00:11.000,0:00:14.506 but it's actually[br]a mathematical treasure trove. 0:00:14.506,0:00:18.654 Indian mathematicians called it [br]the Staircase of Mount Meru. 0:00:18.654,0:00:21.131 In Iran, it's the Khayyam Triangle. 0:00:21.131,0:00:23.738 And in China, it's Yang Hui's Triangle. 0:00:23.738,0:00:28.033 To much of the Western world,[br]it's known as Pascal's Triangle 0:00:28.033,0:00:31.085 after French mathematician Blaise Pascal, 0:00:31.085,0:00:35.234 which seems a bit unfair[br]since he was clearly late to the party, 0:00:35.234,0:00:37.476 but he still had a lot to contribute. 0:00:37.476,0:00:42.270 So what is it about this that has so [br]intrigued mathematicians the world over? 0:00:42.270,0:00:46.124 In short, [br]it's full of patterns and secrets. 0:00:46.124,0:00:49.428 First and foremost, there's the pattern[br]that generates it. 0:00:49.428,0:00:54.477 Start with one and imagine invisible[br]zeros on either side of it. 0:00:54.477,0:00:58.592 Add them together in pairs, [br]and you'll generate the next row. 0:00:58.592,0:01:02.066 Now, do that again, and again. 0:01:02.066,0:01:05.784 Keep going and you'll wind up [br]with something like this, 0:01:05.784,0:01:09.325 though really Pascal's Triangle [br]goes on infinitely. 0:01:09.325,0:01:14.914 Now, each row corresponds to what's called[br]the coefficients of a binomial expansion 0:01:14.914,0:01:18.898 of the form (x+y)^n, 0:01:18.898,0:01:21.307 where n is the number of the row, 0:01:21.307,0:01:23.746 and we start counting from zero. 0:01:23.746,0:01:26.552 So if you make n=2 and expand it, 0:01:26.552,0:01:31.107 you get (x^2) + 2xy + (y^2). 0:01:31.107,0:01:34.023 The coefficients, [br]or numbers in front of the variables, 0:01:34.023,0:01:38.397 are the same as the numbers in that row[br]of Pascal's Triangle. 0:01:38.397,0:01:43.256 You'll see the same thing with n=3,[br]which expands to this. 0:01:43.256,0:01:48.493 So the triangle is a quick and easy way[br]to look up all of these coefficients. 0:01:48.493,0:01:50.037 But there's much more. 0:01:50.037,0:01:52.897 For example, add up [br]the numbers in each row, 0:01:52.897,0:01:56.039 and you'll get successive powers of two. 0:01:56.039,0:02:01.221 Or in a given row, treat each number[br]as part of a decimal expansion. 0:02:01.221,0:02:07.835 In other words, row two is[br](1x1) + (2x10) + (1x100). 0:02:07.835,0:02:12.111 You get 121, which is 11^2. 0:02:12.111,0:02:15.872 And take a look at what happens[br]when you do the same thing to row six. 0:02:15.872,0:02:25.136 It adds up to 1,771,561,[br]which is 11^6, and so on. 0:02:25.136,0:02:27.890 There are also geometric applications. 0:02:27.890,0:02:29.691 Look at the diagonals. 0:02:29.691,0:02:34.117 The first two aren't very interesting:[br]all ones, and then the positive integers, 0:02:34.117,0:02:36.656 also known as natural numbers. 0:02:36.656,0:02:40.707 But the numbers in the next diagonal[br]are called the triangular numbers 0:02:40.707,0:02:42.783 because if you take that many dots, 0:02:42.783,0:02:46.389 you can stack them [br]into equilateral triangles. 0:02:46.389,0:02:49.307 The next diagonal [br]has the tetrahedral numbers 0:02:49.307,0:02:54.622 because similarly, you can stack[br]that many spheres into tetrahedra. 0:02:54.622,0:02:57.996 Or how about this:[br]shade in all of the odd numbers. 0:02:57.996,0:03:00.881 It doesn't look like much[br]when the triangle's small, 0:03:00.881,0:03:03.298 but if you add thousands of rows, 0:03:03.298,0:03:07.439 you get a fractal [br]known as Sierpinski's Triangle. 0:03:07.439,0:03:10.756 This triangle isn't just [br]a mathematical work of art. 0:03:10.756,0:03:12.742 It's also quite useful, 0:03:12.742,0:03:15.481 especially when it comes [br]to probability and calculations 0:03:15.481,0:03:18.566 in the domain of combinatorics. 0:03:18.566,0:03:20.454 Say you want to have five children, 0:03:20.454,0:03:22.270 and would like to know the probability 0:03:22.270,0:03:26.590 of having your dream family [br]of three girls and two boys. 0:03:26.590,0:03:28.388 In the binomial expansion, 0:03:28.388,0:03:32.116 that corresponds [br]to girl plus boy to the fifth power. 0:03:32.116,0:03:33.660 So we look at the row five, 0:03:33.660,0:03:37.131 where the first number [br]corresponds to five girls, 0:03:37.131,0:03:39.929 and the last corresponds to five boys. 0:03:39.929,0:03:42.692 The third number [br]is what we're looking for. 0:03:42.692,0:03:46.642 Ten out of the sum [br]of all the possibilities in the row. 0:03:46.642,0:03:51.490 so 10/32, or 31.25%. 0:03:51.490,0:03:55.316 Or, if you're randomly [br]picking a five-player basketball team 0:03:55.316,0:03:57.084 out of a group of twelve friends, 0:03:57.084,0:04:00.102 how many possible groups [br]of five are there? 0:04:00.102,0:04:05.062 In combinatoric terms, this problem would[br]be phrased as twelve choose five, 0:04:05.062,0:04:07.237 and could be calculated with this formula, 0:04:07.237,0:04:11.708 or you could just look at the sixth [br]element of row twelve on the triangle 0:04:11.708,0:04:13.383 and get your answer. 0:04:13.383,0:04:15.079 The patterns in Pascal's Triangle 0:04:15.079,0:04:19.387 are a testament to the elegantly [br]interwoven fabric of mathematics. 0:04:19.387,0:04:23.271 And it's still revealing fresh secrets[br]to this day. 0:04:23.271,0:04:27.422 For example, mathematicians recently [br]discovered a way to expand it 0:04:27.422,0:04:30.019 to these kinds of polynomials. 0:04:30.019,0:04:31.758 What might we find next? 0:04:31.758,0:04:34.097 Well, that's up to you.