00:01
Suppose we have one factor multiplied by itself a total of five times.
00:04
It's really complicated to expand out this factor over and over again.
00:08
So what i could use is i could use the binomial theorem, right? binomial theorem means k equals 1 to the number n, in choose k, of my first term.
00:19
A is my first term to the n minus k.
00:23
And then my second term, i'm subtracting or note that i'm adding a negative, negative b.
00:30
To the kth term.
00:33
And note, since i'm going to the fifth power, all my ends are replaced with five.
00:40
So i could use this to just the summation to evaluate this.
00:44
I'm going to start with k equals going from zero to n.
00:48
I'm going to do five choose zero of a to the five minus zero times negative b to the zero, plus now i'm going to choose one, a to the five minus one, negative b, negative b, to the 0 to the first power, plus 5 choose 2, a to the 5 minus 2, and then negative b to the second power, and so on and so forth until i reached my go all the way to 5.
01:20
So what i could also do from here is i note that i have a bunch of 5 choose something.
01:26
So what i'm going to do is i'm going to evaluate each binomial coefficient, and note that there's a pattern when i'm evaluating binomial coefficient.
01:33
As well.
01:37
So for example, 5 choose 0, if i were to evaluate this, this would be the same as 5 factorial over the bottom term factorial, 5 minus 0 factorial.
01:48
These 5 factorials would end up canceling out, so this would just be 1.
01:52
And note that i could also write this as 5 factorial over 5 factorial, 5 minus 0 factorial, which is the same as, or 5 minus 5 factorial, which is the same as 5 choose 5...