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College Algebra: Solving Combination Problems

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About this Lesson

  • Type: Video Tutorial
  • Length: 8:07
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 87 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
College Algebra: Further Topics (12 lessons, $17.82)
College Algebra: Combinations and Probability (4 lessons, $4.95)

In this lesson, you will begin by reviewing permutations. Then, you will learn about mathematical combinations and how to evaluate and solve combinatorics problems that involve combinations. Most often, the formula for the binomial coefficient is used to solve these problems. This is also called the choose function and read as ‘n choose k.’ Where order matters in permutations (e.g. A B C is distinct from A C B), combinations do not take sequence into account. Combination problems are things like, How many different sets of 6 numbers are there that can be selected from a broader set of 50 (to give you an idea of the odds that you'll win a Pick 6 Lottery game?

For Professor Burger's lesson on combinatorics problems involving permutations, check out http://www.mindbites.com/lesson/812.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Okay, so we now know how to actually find permutations, that is the different ways the different ways of selecting things from a collection where the order matters. And we saw, in fact, that if we wanted to take a look at all the different ways of placing these three blocks down in a certain order, yellow, blue, green, yellow, blue, green, that there are six. Now, the question is, suppose I just want to pick two of them out, but order doesn’t matter so I don’t care if it’s going to be yellow or blue, it could be blue, yellow, I don’t care, but I’m just picking out this particular grouping. Well then, how many would there be of that? Well, if you think about it, there would be, I just counted twice because blue, yellow is here, but then yellow, blue is over here, so I counted those things twice. Blue, green I counted here, but then green, blue I counted here, and yellow, green, I counted here and green, yellow I counted here. So basically I counted everything twice. So I took this answer of 6 and divided it by 2, that would give me the correct answer of just 3, there are three different ways. There’s this way, there’s this way, and there’s this way. So there’s three ways of picking two things if order doesn’t matter and, in fact, this is exactly how all the lottery games, the state lottery games, are played. You have to select numbers from 1 to 50 or something, but the order that they come out, out of that thing spitting out the ping-pong balls, doesn’t make a difference, all you care about is what the numbers are themselves, order is irrelevant.
So how do you actually figure out the number of ways of choosing things if order doesn’t matter? Well, these are actually called Combinations. So a combination is just the number of ways of selecting things when order does not matter.
So suppose I have a collection of n things and I want to see how many different ways there are for me to select r of them, but order doesn’t make a difference. Well, first of all I’d write this as, c of n choose r, this means I have n things and I want to see how many different ways there are of me picking r of them where order doesn’t make a difference, I don’t care. So I pick blue and then red and that versus red then blue, those are the same things, that’s called a Combination, and this is read, n choose r, n things and I’m going to choose r of them. And what’s the formula? Well, the first thing I do is I look at all the different ways of me just picking r things. Well, that’s the permutation that we just talked about. That’s just n! ÷ n - r!, or if you would read this out loud n ÷ n - r because of the factorial.
But think about it, here with the permutation order matters, now I don’t care about the order, so I want to get rid of all the repetition. Well, how many different ways are there to order those r things I pick out? Well, that was the very first thing we talked about, that’s r factorial. That’s the total number of ways of ordering r things. And I want to get rid of all of them so I’ll just divide this by r factorial, and there’s the formulation, there’s the formula for combinations of n choose r things. And, in fact, you may recognize this as the Binomial Coefficient. And, in fact, this is the formula for the nr binomial coefficient, it’s n choose r. All the different ways in the expansion of an nth degree thing of picking the r terms, but order doesn’t make a difference, I’m just going to add them all up. And so, in fact, that was where that formula came from, you can finally see that.
So let me look at an example. Suppose you have a deck of regular 52 cards. And so, you can see, it’s regular playing cards. Well, suppose you’re being dealt 5 cards but hope for something, a question that’s reasonable to ask is necessary in computing likelihood, is how many different possible hands are there of 5 cards? Now, with the hand, this the 4 of diamonds, the 6 of hearts, the 2 of clubs, the 8 of hearts and the 6 of clubs, of course, order doesn’t matter. To me this hand is the same as that hand, and it’s the same thing as this hand, and it’s the same thing as this hand, it’s just hands. So, in fact, order doesn’t matter, I just want to know how many different ways are there of me picking 5 cards out of 52.
Well, let’s see. That would just be 52 choose 5, and so that would be c of 52, 5, 52 things, I want to choose 5 but order doesn’t matter. So that’s going to be 52! ÷, and I have r, r is 5, so 5! x n - r. So that’s going to be what? That’s going to be 52 - 5, which is going to be 47 factorial. Okay, now what does that equal? Well, you can plug into a calculator or we can simplify that a bit. You can simply it a bit if you can write out a few of these terms here, so it’s going to be 52 x 51 x 50 x 49 x 48, and then also put in x 47 factorial because that’s the rest of it, 47, 46, 45, and so forth. Then on the bottom I have 5 x 4 x 3 x 2 x 1, but now I just have x 47 factorial. And what you notice is the 47 factorial’s been cancelled. So all I have to do, it’s still a lot, is take 52 x 51 x 50 x 49 x 48 ÷ 5 x 4 x 3 x 2 x 1, and if you work that out on a calculator, what you see is 2,598,960. That is the total number of different possible deals of 5 you can possibly get from a 52 card regulation playing deck and I just found it by looking at the combinations.
So, now what I thought I would do is just end, in fact, by actually bringing one of my distinguished colleagues in the mathematical community to give his commentary on this kind of thing because he’s an expert, he’s a Combinataurus, in fact, an expert on this card counting thing and so forth. So, to recap and summarize this, I want to bring in this expert. So please welcome a special guest lecture appearance by the Punching Monkey. Please welcome him. Thank you very much.
[Punching Monkey skit]

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