1
00:00:07,603 --> 00:00:11,000
This may look like a neatly arranged
stack of numbers,

2
00:00:11,000 --> 00:00:14,506
but it's actually
a mathematical treasure trove.

3
00:00:14,506 --> 00:00:18,654
Indian mathematicians called it 
the Staircase of Mount Meru.

4
00:00:18,654 --> 00:00:21,131
In Iran, it's the Khayyam Triangle.

5
00:00:21,131 --> 00:00:23,738
And in China, it's Yang Hui's Triangle.

6
00:00:23,738 --> 00:00:28,033
To much of the Western world,
it's known as Pascal's Triangle

7
00:00:28,033 --> 00:00:31,085
after French mathematician Blaise Pascal,

8
00:00:31,085 --> 00:00:35,234
which seems a bit unfair
since he was clearly late to the party,

9
00:00:35,234 --> 00:00:37,476
but he still had a lot to contribute.

10
00:00:37,476 --> 00:00:42,270
So what is it about this that has so 
intrigued mathematicians the world over?

11
00:00:42,270 --> 00:00:46,124
In short, 
it's full of patterns and secrets.

12
00:00:46,124 --> 00:00:49,428
First and foremost, there's the pattern
that generates it.

13
00:00:49,428 --> 00:00:54,477
Start with one and imagine invisible
zeros on either side of it.

14
00:00:54,477 --> 00:00:58,592
Add them together in pairs, 
and you'll generate the next row.

15
00:00:58,592 --> 00:01:02,066
Now, do that again and again.

16
00:01:02,066 --> 00:01:05,784
Keep going and you'll wind up 
with something like this,

17
00:01:05,784 --> 00:01:09,325
though really Pascal's Triangle 
goes on infinitely.

18
00:01:09,325 --> 00:01:14,914
Now, each row corresponds to what's called
the coefficients of a binomial expansion

19
00:01:14,914 --> 00:01:18,898
of the form (x+y)^n,

20
00:01:18,898 --> 00:01:21,307
where n is the number of the row,

21
00:01:21,307 --> 00:01:23,746
and we start counting from zero.

22
00:01:23,746 --> 00:01:26,552
So if you make n=2 and expand it,

23
00:01:26,552 --> 00:01:31,107
you get (x^2) + 2xy + (y^2).

24
00:01:31,107 --> 00:01:34,023
The coefficients, 
or numbers in front of the variables,

25
00:01:34,023 --> 00:01:38,397
are the same as the numbers in that row
of Pascal's Triangle.

26
00:01:38,397 --> 00:01:43,256
You'll see the same thing with n=3,
which expands to this.

27
00:01:43,256 --> 00:01:48,493
So the triangle is a quick and easy way
to look up all of these coefficients.

28
00:01:48,493 --> 00:01:50,037
But there's much more.

29
00:01:50,037 --> 00:01:52,897
For example, add up 
the numbers in each row,

30
00:01:52,897 --> 00:01:56,039
and you'll get successive powers of two.

31
00:01:56,039 --> 00:02:01,221
Or in a given row, treat each number
as part of a decimal expansion.

32
00:02:01,221 --> 00:02:07,835
In other words, row two is
(1x1) + (2x10) + (1x100).

33
00:02:07,835 --> 00:02:12,111
You get 121, which is 11^2.

34
00:02:12,111 --> 00:02:15,872
And take a look at what happens
when you do the same thing to row six.

35
00:02:15,872 --> 00:02:25,136
It adds up to 1,771,561,
which is 11^6, and so on.

36
00:02:25,136 --> 00:02:27,890
There are also geometric applications.

37
00:02:27,890 --> 00:02:29,691
Look at the diagonals.

38
00:02:29,691 --> 00:02:34,117
The first two aren't very interesting:
all ones, and then the positive integers,

39
00:02:34,117 --> 00:02:36,656
also known as natural numbers.

40
00:02:36,656 --> 00:02:40,707
But the numbers in the next diagonal
are called the triangular numbers

41
00:02:40,707 --> 00:02:42,783
because if you take that many dots,

42
00:02:42,783 --> 00:02:46,389
you can stack them 
into equilateral triangles.

43
00:02:46,389 --> 00:02:49,307
The next diagonal 
has the tetrahedral numbers

44
00:02:49,307 --> 00:02:54,622
because similarly, you can stack
that many spheres into tetrahedra.

45
00:02:54,622 --> 00:02:57,996
Or how about this:
shade in all of the odd numbers.

46
00:02:57,996 --> 00:03:00,881
It doesn't look like much
when the triangle's small,

47
00:03:00,881 --> 00:03:03,298
but if you add thousands of rows,

48
00:03:03,298 --> 00:03:07,439
you get a fractal 
known as Sierpinski's Triangle.

49
00:03:07,439 --> 00:03:10,756
This triangle isn't just 
a mathematical work of art.

50
00:03:10,756 --> 00:03:12,742
It's also quite useful,

51
00:03:12,742 --> 00:03:15,481
especially when it comes 
to probability and calculations

52
00:03:15,481 --> 00:03:18,566
in the domain of combinatorics.

53
00:03:18,566 --> 00:03:20,454
Say you want to have five children,

54
00:03:20,454 --> 00:03:22,270
and would like to know the probability

55
00:03:22,270 --> 00:03:26,590
of having your dream family 
of three girls and two boys.

56
00:03:26,590 --> 00:03:28,388
In the binomial expansion,

57
00:03:28,388 --> 00:03:32,116
that corresponds 
to girl plus boy to the fifth power.

58
00:03:32,116 --> 00:03:33,660
So we look at the row five,

59
00:03:33,660 --> 00:03:37,131
where the first number 
corresponds to five girls,

60
00:03:37,131 --> 00:03:39,929
and the last corresponds to five boys.

61
00:03:39,929 --> 00:03:42,692
The third number 
is what we're looking for.

62
00:03:42,692 --> 00:03:46,642
Ten out of the sum 
of all the possibilities in the row.

63
00:03:46,642 --> 00:03:51,490
so 10/32, or 31.25%.

64
00:03:51,490 --> 00:03:55,316
Or, if you're randomly 
picking a five-player basketball team

65
00:03:55,316 --> 00:03:57,084
out of a group of twelve friends,

66
00:03:57,084 --> 00:04:00,102
how many possible groups 
of five are there?

67
00:04:00,102 --> 00:04:05,062
In combinatoric terms, this problem would
be phrased as twelve choose five,

68
00:04:05,062 --> 00:04:07,237
and could be calculated with this formula,

69
00:04:07,237 --> 00:04:11,708
or you could just look at the sixth 
element of row twelve on the triangle

70
00:04:11,708 --> 00:04:13,383
and get your answer.

71
00:04:13,383 --> 00:04:15,079
The patterns in Pascal's Triangle

72
00:04:15,079 --> 00:04:19,387
are a testament to the elegantly 
interwoven fabric of mathematics.

73
00:04:19,387 --> 00:04:23,271
And it's still revealing fresh secrets
to this day.

74
00:04:23,271 --> 00:04:27,422
For example, mathematicians recently 
discovered a way to expand it

75
00:04:27,422 --> 00:04:30,019
to these kinds of polynomials.

76
00:04:30,019 --> 00:04:31,758
What might we find next?

77
00:04:31,758 --> 00:04:34,097
Well, that's up to you.