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Binomial Theorem - Problem 1
Combinations or choosing. So this is an application of factorials that come up in algebra 2 and as well that come up in pre-calculus. But for right now we’re going to focus on the basic premise of choosing, that’s the definition of it. And what this is saying is n choose r, and what it comes with is this weird looking formula, okay?
And factorial, so it’s the number on the outside, factorial, divided by the difference of your two numbers factorial times the other number factorial. That’s the definition, sort of confusing on the surface but we’ll look into it a little bit more and also one thing you should know is that this is sometimes written also as parenthesis n and r. So it’s just basically n choose r. It’s a different way of writing the exact same formula we have over here.
So let’s actually put this into application. So what we have here is 9 choose 3. Putting this into our equation, what we end up having is the n factorial is the 9, this becomes 9 factorial. The n minus r so the big number minus the small number becomes 9 minus 3, this becomes 9 minus 3 factorial and lastly the r factorial, the small number factorial is just 3 factorial.
Simplifying this up, we know that we have 9 factorial and then the 9 minus 3 is just 6 factorial, 3 factorial. A factorial remember is just that number multiplied by every number smaller than that, so what we have here is 9 times 8 times 7 and so on and so forth. This is 6 times 5 times 4, so the 6 in down all cancel leaving us with a 9 times 8 times 7 in the numerator, the 6 and down all cancelled out, over 3 factorial, which is just 3 times 2 times 1. Again that 1 we don’t really need because it’s just times 1 but it’s always good to throw it in there.
Now, we just have a fraction which we can cancel terms and try to make our life a little bit easy. 9 over 3 goes to 3, 8 over 2 ends up with a 4. So what we end up with is 3 times 4 times 7 and we have nothing left in the bottom. 3 times 4 is 12, 12 times 7 is, let’s make sure we get it right, 12 times 7 is 84. So using our formula and definition, we found that 9 choose 3 is 84. So it’s a scary-looking formula but as long as you know where each term goes, it’s pretty straight forward to use.