question from Manhattan online article about Combinatorics

[wignewton]

The article I am referencing can be found at http://www.manhattangmat.com/strategy-series-probability.cfm. My question comes from the Section titled "Formulaic Approach" 9th paragraph. I was understanding this line method that you introduce in the article completely until this section. Suddenly Factorial 2 (2!) appears as the denominator in the equation when there are 5 people to fill 3 spaces. It is 6 in the morning and I haven`t had my coffee, so maybe I am not reading closely, but is there an explanation for this being added to the equation? BTW, I have just bought the Word Translations book so I can get a better understanding of these kinds of problems. Yours truly,
Mathdumb

[JonathanSchneider]

The factorial appears there because the numerator of that fraction is 5!. Of course, as you can see from the slot method presented just before, we only had 5*4*3. We did NOT have the full 5! (which would include ...*2*1). The next step (the one that you referenced) simply shows 5*4*3 as 5!/2!, showing that the 2! has somehow been canceled out. (We later go into why that is the case.)

You'll get more exposure to these problems later on in the course + books. However, keep in mind that they are NOT the most commonly tested material. So, don't overstudy for these at the expense of other topics.