# A classical problem in combinatorics/probability

I read this problem in Cognition and Chance by Raymond Nickerson (the problem is stated not discussed)

A bag has 2n balls, two of which are marked '1', another two marked '2' and so
on. m balls are chosen, find the probability that k pairs are still in the bag.


Here's my hand at the solution:

The $k$ pairs are chosen by $n \choose k$. And let $x_i$ be the number of balls chosen of type $i$ (excluding the $k$ previously chosen) and these $x_i$'s should satisfy: $$x_1 + x_2 + \cdots + x_{n-k} = m .\tag{1}$$ and clearly $$1 \leq x_i \leq 2 \tag{2}$$