Still remember the binomial theorem? It’s the formula for generating the coefficients of a “binomial”–a fancy word for any two-term piece of algebra e.g. (1 + x) or (a + b)–raised to any power you like. As it happens, the coefficients turn out to be the numbers in the appropriate row of Pascal’s triangle. For example if we square the following binomial . . .
(x + y)2 = x2 + 2xy + y2
. . . we get the coefficients 1 2 1.
Can you see that these match the numbers in third row of Pascal’s triangle? It works for whatever power you care to raise a binomial to. For a cube, go to the fourth row, For a quartic, go to the fifth row. And so on.
So where do these numbers come from? Short answer: combinatorics. If that makes everything crystal clear then you’re gravy. Move along. For everyone else, consider the expansion of a quartic:
(x + y)4 = (x + y)(x + y)(x + y)(x + y)
If you think about it, the value of the coefficient simply reflects the number of ways in which you can make that particular combination from the binomials available. So, the only way to produce the x4 term is to pick an x from each bracket–and there’s only one way you can do that. Same idea goes for the y4 term. The term x3y is more tricky. Now we have to pick three xs and one y from the four brackets. There are four ways to do this (xxxy, xxyx, xyxx, yxxx). In mathematical language we say that from a set of four (binomials) we are choosing three (xs); 4C3. For the next term x2y2, the coefficient will be 4C2. Lo-and-behold we know why the numbers work!
Anyway, back to the question I posed in the title, aside from calculating the coefficients of binomial expansions, what use is the theorem?
Well, quite a lot actually. The binomial theorem can be leveraged in any situation where you have an event with fixed odds repeated a set number of times. Let’s say you’re a car manufacturer who also runs a network of garages for drivers of your vehicles. Imagine on any given day there’s a 0.1% chance one of your vehicles will break down and need repair. How many garages should you run? Too many and you’ve got mechanics sitting idle–and getting paid for the privilege. Too few and you’ve got angry customers unable to get their cars fixed.
Of course, the answer depends on the number of your vehicles on the road. Let’s say there’s ten thousand. Binomial theorem lets you calculate the chance of a given number of vehicles breaking down–and therefore allows you to make informed choices about how many garages you should provide (the starting point for the calculation is (0.001 + 0.999)10000). That could make it pretty smart business.
This is a concept that turns up in everything from weather forecasting to casino gambling, making the binomial theorem very useful indeed.
Do you know any other uses for the binomial theorem?
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