Is there a resource (book/online) that shows how many ways an opening can be reached?
The point from chubbycantorset's answer about the futility of seeking the number of all ways to reach a position is a good one. Scid has a way of offering more limited (and more practically useful anyway) information than that. Its Opening Report tool allows you to input a position, and then generate a comprehensive statistical report covering games that reached that position (in whatever database you have open). Among the data presented is a tally of how many move-orders were used to reach the position in question that appeared in actual games from the database, as well as a list of the most popular.
For instance, in my big database of millions of games, when I generate a report for the standard Sicilian Scheveningen position
[fen "rnbqkb1r/pp3ppp/3ppn2/8/3NP3/2N5/PPP2PPP/R1BQKB1R w KQkq - 0 1"]
part of the information I receive is as follows, where the numbers in parentheses indicate how many games feature the move-order in question:
There were 82 move orders reaching this position. The top 8 are: 1: 1. e4 c5 2. Nf3 e6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 d6 (18538) 2: 1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 e6 (11028) 3: 1. e4 c5 2. Nf3 d6 3. d4 Nf6 4. Nc3 cxd4 5. Nxd4 e6 (646) 4: 1. e4 c5 2. Nf3 e6 3. Nc3 d6 4. d4 cxd4 5. Nxd4 Nf6 (597) 5: 1. e4 c5 2. Nc3 e6 3. Nf3 d6 4. d4 cxd4 5. Nxd4 Nf6 (258) 6: 1. e4 c5 2. Nf3 e6 3. d4 cxd4 4. Nxd4 d6 5. Nc3 Nf6 (144) 7: 1. e4 e6 2. d4 c5 3. Nf3 cxd4 4. Nxd4 Nf6 5. Nc3 d6 (117) 8: 1. e4 c5 2. Nf3 d6 3. Nc3 Nf6 4. d4 cxd4 5. Nxd4 e6 (92)
I don't think so, because, for example, the french defence can be reached in the following way:
1.Nf3 Nf6 2.Ng1 Ng8 3.Nc3 Nc6 4.Nb1 Nb8 5. e4 e6. Of course, the knight moves can be repeated more times, with different combinations. Also,
e4 can be delayed with
e3, when black "wastes" a move with a knight. Thus, it's not too useful to have such a resource.