Evaluate 133 choose two.
In order to actually evaluate a combination in this form, we have this formula that tells us that if we have 𝑛 choose 𝑟, this is equal to 𝑛 factorial divided by 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. It’s worth reminding ourselves at this point what a factorial is.
Well, if we have 𝑛 factorial, what this means is the product of all positive integers that are less than or equal to 𝑛. For example, if we had five factorial, this will be equal to five multiplied by four multiplied by three multiplied by two multiplied by one, because these are all the positive integers that are less than or equal to five.
Okay, great! So now let’s use this to actually evaluate 133 choose two. So we’re gonna have 133 choose two is equal to 133 factorial over two factorial multiplied by 133 minus two factorial. So then if we actually simplify this, what we’re gonna get is 133 factorial over two factorial multiplied by 131 factorial. So let’s think about what this actually means.
Well, it means that our numerator is gonna be 133 factorial, which is 133 times 132 times 131 times 130 etc. And then our denominator’s gonna be two multiplied by one, because it’s two factorial, and then multiplied by 131 times 130 times 129 etc. because that’s gonna be 131 factorial.
So therefore, if we actually divide through by 131 factorial, we’re gonna be left with 133 times 132 over two times one, which is gonna give us 133 multiplied by 132 over two, which is equal to 17556 over two. So therefore, we can say that 133 choose two is equal to 8778. And that’s because we had 17556 divided by two.