This week Occupy Math is championing an item of mathematics that is underrepresented in secondary education – the choice numbers or, in techno-speak, the binomial coefficients. These numbers let you compute the number of ways to choose some, all, or none of the members of a set. They show up all over the place in gambling and probability theory. They were discovered by Blaise Pascal, whose picture appears above, initially to understand gambling, but in the end because they contain a cornucopia of patterns.
The choice numbers are not hard to understand, but Occupy Math has heard public school teachers object that “that’s not mathematics!” when they see them.
Suppose we have five desserts available (actual desserts, not types of desserts) and are taking three of them back to the table for yourself and two friends. If these deserts are (1) apple pie, (2) cherry pie, (3) chocolate cake, (4) carrot cake, and (5) butterscotch pudding, then the possible choices of three desserts are: 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345. There are ten such choices – and the systematic, numerical listing Occupy Math used makes it fairly easy to be sure that the list of ten choices supplied is complete. What if there were sixteen deserts and five people at your table? Then the number of choices is 4,368. This would make a listing way too much work. Fortunately, Pascal found a beautiful pattern (and a less beautiful formula).
To make a new row of Pascal’s Triangle you put in bordering ones flanking the new row and fill in between them by adding the entries above left and above right of the position youíre filling. Many thanks to Wikipedia for the animated triangle! Now we need to figure out what it means.
The row with a number N in the second position contains the choice numbers for a set of N objects.
The row 1 4 6 4 1, for example, tells us that there is one way to choose no objects out of four, four ways to choose one object, six ways to choose two objects, and so on. Our original example shows there are ten ways to choose three objects out of five – let’s see if ten appears in the correct position in a larger version of the triangle. Since the first entry is for choosing no objects, the fourth entry in the row starting “1 5 …” is the number of ways to choose three of five objects – and sure enough, that entry is 10. It’s also light cyan, to make it easy to find.
Alert readers will have noticed that the triangle is left-right symmetric. This follows from the rather Zen observation that choosing some members of a set can be done in the same number of ways as not choosing the other members of the set. In other words, to choose three desserts is to not choose two desserts (at least when five are available). This left-right symmetry is just one of many patterns in the triangle.
Another source of patterns is to look at where even numbers (blue), multiples of three (green), and multiples of five (red) appear. If we blend these three colorings we get:
Now Occupy Math presents the formula, which is based on factorials. The factorial of a (positive, whole) number is the result of multiplying the number by all the smaller whole numbers down to one. Factorials are denoted with an exclamation mark so:
… and so on. Occupy Math hopes that factorials are known to many of his readers. Once, when teaching a third year university course, Occupy Math had an otherwise excellent student that had substantial trouble with factorials. Occupy Math wrote “5” on the board and said “Please say this number out loud.” The student said “five”. Occupy Math then made the same request for “5!” and the student said “FIVE!”, rather loudly. He was not familiar with factorials.
The way we write choice numbers is nCr for “number of ways to pick r members out of a set with n members”. The number is spoken “n choose r”. The formula is:
nCr=n!/(r! x (n-r)!)
and is often used when we are too far down the triangle to do all the rows and only one choice number is needed. Let’s work the formula on our three-of-five-desserts example. 5C3=5!/(3!x2!)=120/(2×6)=120/12=10, so the formula works on the example. We would say “five choose three is ten”.
The choice numbers show up in algebra and probability as binomial coefficients and are useful for understanding things like flipped coins and radiation counts, as well as being useful for speeding up arithmetic on polynomials. Occupy Math is disappointed that these fundamental numbers aren’t part of what is taught in many high schools.
Pascal first looked at these numbers for gambling – so let’s try an example. If we draw five cards, what’s the chance we get a pair? Well, there are 13 kinds of cards we could have a pair of; we need to choose 2 of the 4 cards of that type; then we need to choose three different other types of cards with four choices of suit for each. So, since independent choices are multiplied, the number of ways to get a pair is:
This is not a probability – it is a count. So now we divide by the total number of five card draws, the number of ways to choose five cards from fifty-two cards or 52C5=2,598,960, and we get 1098240/2598960 or about 42.3%.
It’s a little surprising that the odds of drawing a pair are just below 50%
The choice numbers appear in figuring the odds for a majority of types of gambling and, not surprisingly, all over the science of probability. They appear in algebra and the sciences that use algebra, like physics. They also have thousands of patterns beyond the colorful examples provided above. Occupy Math hopes that you’ve managed to stick with us this far in this relatively technical post. Do you have stories of times where a better grip on probability and chance would have helped? Please comment or tweet to share them!
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics