there are about 2500 stations on the national rail network so even if there was just one route and one ticket type and one class of travel for each possible combination that would be 6.25 million tickets
Agreed.
It's half that, if A to B is the same as B to A
Unfortunately A to B is not always the same as B to A, so we have to treat them separately. Sometimes the fare is different.
LONDON TERMINALS - WOKING SDR ANYTIME DAY RETURN £18.60
WOKING - LONDON TERMINALS SDR ANYTIME DAY RETURN £19.40
Sometimes the time restriction is different.
STOCKPORT - SHEFFIELD Route DIRECT CDR OFF-PEAK DAY RETURN £16.60
OUTWARD TRAVEL: By any train except those timed to depart Mondays to Fridays before 0915
RETURN TRAVEL: By any train
SHEFFIELD - STOCKPORT Route DIRECT CDR OFF-PEAK DAY RETURN £16.60
OUTWARD TRAVEL: By any train except those timed to depart Mondays to Fridays before 0930
RETURN TRAVEL: By any train
Sometimes both fare and time restriction are different.
MIDDLESBROUGH - LEEDS SVR OFF-PEAK RETURN £33.90
MONDAYS - FRIDAYS (By any train on other days).
OUTWARD TRAVEL: All trains are barred which are due to arrive in Leeds before 0930
RETURN TRAVEL: By any train.
LEEDS - MIDDLESBROUGH SVR OFF-PEAK RETURN £35.70
Available as listed below on any day.
OUTWARD TRAVEL: Valid on any train. Break of journey is not permitted except to change trains at an intermediate station or to access station facilities.
RETURN TRAVEL: Valid on any train.
I am intrigued where the numbers in the region of 300 million, let alone 3 billion, come from.
Even the lower of these numbers implies that there are, on average, about 100 different ticket types / routes / whatever confounding factors you like that lead to a distinct "fare", for each and every station pair in the country. That sounds tough to believe.
An average of 100 different fares per pair of stations does not appear an unrealistic estimate to me. Taking my most recent long distance journey,
Morley - Bellgrove, there are 14 different First Class tickets and 21 different Standard tickets, a total of 35 different tickets. For each of these 35 tickets there may be a discount of:
0% - Adult
2% to 11% - Some fares routed EC & CONNECTNS if purchased via
East Coast
34% - Railcard
36% to 41% - Railcard with some fares routed EC & CONNECTNS if purchased via
East Coast
50% - Child
75% - Some fares with staff discount card.
81% - Child with some Railcard holders
88% - Some Child fares for dependant of person with staff discount card.
and probably some others I've forgotten. This gives more than 200 different fares between a station on the edge of Leeds and a Glasgow suburban station. My guess is that the total number of Morley - Bellgrove tickets sold each year is under 200.
If we assume an "s" at the end of 'combination', and that your interest depends on that word 'combinations', then I will guess that the possible and available combinations are likely to be an order of magnitude greater than merely the number of available tickets. i.e. 3 billion.
I think 3 billion may be an underestimate. Using the
Morley - Bellgrove example, there are
39 intermediate stations on the shortest route by rail. Some ticket types may be split at any or all of these stations, or at any of several hundred stations on other Permitted Routes
Morley - Bellgrove. My guess is that there are more than 30 billion possible combinations of fares.