1
00:00:00,306 --> 00:00:01,680
PROFESSOR: Two
goals really-- one

2
00:00:01,680 --> 00:00:05,074
is to introduce
students to an area

3
00:00:05,074 --> 00:00:07,740
of mathematics that really isn't
covered much in the high school

4
00:00:07,740 --> 00:00:09,390
curriculum.

5
00:00:09,390 --> 00:00:11,550
I think there's a lot
of fascinating problems

6
00:00:11,550 --> 00:00:14,580
in combinatorics and
probability that kids

7
00:00:14,580 --> 00:00:18,750
don't see much when they're
in their earlier studies,

8
00:00:18,750 --> 00:00:21,750
and they're extremely relevant.

9
00:00:21,750 --> 00:00:24,690
I guess there's sort of three
main reasons I give students

10
00:00:24,690 --> 00:00:28,320
at the first day of the
course as to why combinatorics

11
00:00:28,320 --> 00:00:30,010
is an interesting subject.

12
00:00:30,010 --> 00:00:32,444
The first is something that
I guess has always been true,

13
00:00:32,444 --> 00:00:34,110
but we've only really
become aware of it

14
00:00:34,110 --> 00:00:35,693
in the last 50 years--
is the universe

15
00:00:35,693 --> 00:00:38,280
is sort of fundamentally
discrete and digital,

16
00:00:38,280 --> 00:00:40,440
whether you're talking
about quarks inside atoms,

17
00:00:40,440 --> 00:00:42,690
or electrons in
orbits around atoms,

18
00:00:42,690 --> 00:00:46,549
or the way molecules are
built, or the DNA in our cells.

19
00:00:46,549 --> 00:00:48,840
All of these things are made
up of discrete structures,

20
00:00:48,840 --> 00:00:51,550
and to understand how they
work and how they interact,

21
00:00:51,550 --> 00:00:54,810
we need to be able to understand
how these pieces fit together.

22
00:00:54,810 --> 00:00:56,730
So whether you're
a physicist trying

23
00:00:56,730 --> 00:00:58,624
to understand subatomic
particle collisions

24
00:00:58,624 --> 00:01:00,540
or a chemist trying to
understand the patterns

25
00:01:00,540 --> 00:01:05,340
in the periodic table,
combinatorics and combinations

26
00:01:05,340 --> 00:01:10,250
of objects are an important
part of that process.

27
00:01:10,250 --> 00:01:13,100
Building on that, and one of
the major themes of the course,

28
00:01:13,100 --> 00:01:16,220
is probability theory,
and sort of any question

29
00:01:16,220 --> 00:01:19,640
about probability is really two
questions about counting-- sort

30
00:01:19,640 --> 00:01:21,860
of how many things
could possibly happen,

31
00:01:21,860 --> 00:01:24,050
and how many of those
things correspond to what

32
00:01:24,050 --> 00:01:25,500
you're interested in?

33
00:01:25,500 --> 00:01:27,812
And if I were to pick out
one piece of mathematics

34
00:01:27,812 --> 00:01:29,270
that I think
everybody should know,

35
00:01:29,270 --> 00:01:31,647
whether they're
interested in math or not,

36
00:01:31,647 --> 00:01:33,980
it would be sort of a basic
understanding of probability

37
00:01:33,980 --> 00:01:34,340
theory.

38
00:01:34,340 --> 00:01:35,881
I mean, we spent a
lot of time making

39
00:01:35,881 --> 00:01:39,710
kids memorize the quadratic
formula, and learn trig ratios,

40
00:01:39,710 --> 00:01:44,066
and various other miscellaneous
material that they are probably

41
00:01:44,066 --> 00:01:45,940
not that likely to use
in their everyday life

42
00:01:45,940 --> 00:01:47,330
if they don't end up
going into mathematics.

43
00:01:47,330 --> 00:01:50,390
But probability theory is pretty
fundamental to the decisions we

44
00:01:50,390 --> 00:01:54,170
make everyday in our lives,
especially big decisions

45
00:01:54,170 --> 00:01:57,380
like how we invest our money,
or decisions about our health,

46
00:01:57,380 --> 00:02:00,470
or other decisions that have
a big impact on how we live.

47
00:02:00,470 --> 00:02:03,920
And a lot of people either
never learn probability

48
00:02:03,920 --> 00:02:06,560
or are very confused
by it, and I

49
00:02:06,560 --> 00:02:10,465
think that's an important
skill for people to learn.

50
00:02:10,465 --> 00:02:11,840
And then sort of
the third thing,

51
00:02:11,840 --> 00:02:13,920
for the people who are
interested in mathematics

52
00:02:13,920 --> 00:02:17,150
and combinatorics-- which
tends to be the people who come

53
00:02:17,150 --> 00:02:18,830
to MIT to take these courses--

54
00:02:18,830 --> 00:02:23,660
is to expose them to some
interesting and challenging

55
00:02:23,660 --> 00:02:26,536
problems that, I guess,
open up some areas

56
00:02:26,536 --> 00:02:28,910
of the mathematical universe
that they may not have known

57
00:02:28,910 --> 00:02:30,590
existed.

58
00:02:30,590 --> 00:02:32,595
A lot of the students
that I see are, you know,

59
00:02:32,595 --> 00:02:34,220
kids who were at the
top of their class

60
00:02:34,220 --> 00:02:35,780
or who were bored in
their math classes.

61
00:02:35,780 --> 00:02:37,040
They know everything
that's in the textbook,

62
00:02:37,040 --> 00:02:38,510
and I had much the
same experience

63
00:02:38,510 --> 00:02:40,980
when I was in high school.

64
00:02:40,980 --> 00:02:42,760
And it's pretty
easy to go through--

65
00:02:42,760 --> 00:02:45,410
there's sort of a linear
progression of math courses.

66
00:02:45,410 --> 00:02:47,580
You know, take algebra,
geometry, trigonometry,

67
00:02:47,580 --> 00:02:49,974
calculus-- and view it as
sort of a linear sequence,

68
00:02:49,974 --> 00:02:52,640
and think that eventually you'll
get to the end of that sequence

69
00:02:52,640 --> 00:02:53,931
and know what there is to know.

70
00:02:53,931 --> 00:02:57,342
But in fact, it's a
much broader subject,

71
00:02:57,342 --> 00:02:59,300
and there's a huge amount
of material out there

72
00:02:59,300 --> 00:03:01,500
that is very accessible
to high school students

73
00:03:01,500 --> 00:03:03,740
but is never incorporated
in the curriculum.

74
00:03:03,740 --> 00:03:06,380
I didn't really learn
any serious combinatorics

75
00:03:06,380 --> 00:03:10,250
until I was a sophomore
or even a junior at MIT,

76
00:03:10,250 --> 00:03:11,750
but I found the
subject fascinating.

77
00:03:11,750 --> 00:03:15,910
And I think it's very accessible
and useful to younger students.