The extensions of a telephone switchboard have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, one either gets the same number or a number of an extension that is not in use. What is the maximum possible number of extensions in use?
(a) Less than 45
(c) Between 45 and 55
(e) 55
(b) 45
(d) More than 55
The correct option is (e).

There are two types of extensions that may be in use. We have extensions with

- the two digits being the same $(00,11,22,33,44,55,66,77,88$ and 99 ), for a total of 10, and those with
- the two digits being different. In this case, we have $10 \times 9=90$ numbers, and half of them can be used.

Therefore, the maximum number of extensions in use is $10+45=55$.