information:classification, structural:reformatting
Summation formula and approximation

The telephone numbers may be expressed exactly as a summation

In each term of the first sum, k {\displaystyle k} gives the number of matched pairs, the binomial coefficient (n 2 k) {\displaystyle {\tbinom {n}{2k}}} counts the number of ways of choosing the 2 k {\displaystyle 2k} elements to be matched, and the double factorial

is the product of the odd integers up to its argument and counts the number of ways of completely matching the 2 k selected elements. It follows from the summation formula and Stirling's approximation that, asymptotically,