How to type binomial coefficient $\binom{n}{k}$

The possibility to insert operators and functions as you know them from mathematics is not possible for all things. Usually, you find the special input possibilities on the reference page of the function in the Details section. See for instance the documentation of Integrate.

For Binomial there seems to be no such 2d input, because as you already found out, $\binom{n}{k}$ is interpreted as vector.

The 2d input should in general be used with care, because it is sometimes harder to spot errors. And, when you want to copy a nice looking

into a text editor or to this site, you end up with

\[Integral]Sqrt[x + Sqrt[x]] \[DifferentialD]x

which just looks awful.

Edit

Szabolcs came up with the idea that you could make a button for your needs. To see how this is derived, you first look at the TraditionalForm of Binomial

Binomial[n, k] // TraditionalForm

Now you go in the output cell and navigate to Cell->Show Expression

Cell[BoxData[
 FormBox[
  TemplateBox[{"n","k"},
   "Binomial"], TraditionalForm]], "Output",
 CellChangeTimes->{3.5952851298543243`*^9, 3.595285236592402*^9}]

At this point, you see the you need to create the inner TemplateBox and therefore

DisplayForm[
 ButtonBox[
  TemplateBox[{"\[SelectionPlaceholder]", "\[SelectionPlaceholder]"}, 
   "Binomial"], BaseStyle -> "Paste"]]

creates a button which lets you insert Binomial as correct 2d expression.

I would use the menu option Cell | ConvertTo and select TraditionalForm. The keyboard shortcut for this in OSX is Shift + Command + T. Here is an animation showing this in action: