Why do I need another pair of curly braces?

Question

I have an error in this line of LaTeX.

\frac{ {{2}\choose{2}} {{1}\choose{1}} } { {1}\choose{1} }

The error notes

Ambiguous; you need another { and }.

I matched up the curly braces, and I don't think I need one though. What is going on?

Answer 1

The \choose macro needs to know where the top and the bottom part starts and ends:

\frac{ {2\choose 2} {1\choose 1}}{{1 \choose 1}}

But, as far as I know, \choose is a plainTeX macro, you better use \binom (it is in the same family like \frac but needs the amsmath package):

\frac{\binom{2}{2} \binom{1}{1}}{\binom{1}{1}}

You have also \tbinom and \dbinom for textstyle and displaystyle respectively.

Explanation

Consider the following example:

{1 + 23 \choose 4a - 7} \neq 1 + {23 \choose 4}a - 7

It does render as

A additional {23} and {4} just groups those parts with no relation to the \choose macro. (Kind of like \bfseries does in text mode.)

Answer 2

To add to Qrrbrbirlbel's practical answer, I'd like to explain why exactly TeX says that your expression

\frac{ {{2}\choose{2}} {{1}\choose{1}} } { {1}\choose{1} }

is ambiguous although it doesn't look ambiguous. For simplicity let's look at the simpler expression \frac{1}{ {1}\choose{1} }. By LaTeX's definition of \frac, this expands to

{\begingroup 1 \endgroup \over {1}\choose{1} }

And here you already see the trouble: TeX has no way to know what is meant,

1. \begingroup 1 \endgroup over {1}\choose{1}  or
2. \begingroup 1 \endgroup \over {1} choose {1} .

The braces and \begin/endgroup are not relevant here, so essentially the question is:
1 over 1 \choose 1, or 1 \over 1 choose 1?