Which one of the following is correct? and which one is idiomatic?
This is because a proper education will enhance the chances of women reaching high-level positions, consequently, the percentage of female senior managers will increase eventually.
This because a proper education will enhance women's chances of reaching high-level positions................
This because a proper education will enhance the chances of women to reach high-level positions................
The three phrases:
are all correct. The first two are equivalent, while the third has a similar but not exactly the same meaning. "Chances of reaching" (#2 and equivalently #1) describes a probability of a random event, or statistical frequency. "Chance to reach" (#3) describes an opportunity, or a path that may be pursued by choice. (More on this difference)
The difference between the first two sentences seems to be a matter of style or preferred emphasis. The third one may be more adequate semantically due to the beginning of the sentence - education does indeed create more opportunity, so it enhances women's chances to reach (...) (and statistically it increases the chances of them reaching (...) - it is a subtle difference).
The first suggested sentence seems incorrect in another aspect - using "consequently" as a conjunction. Compound sentences with "and consequently" are more common (see here and here for examples). Alternatively, "consequently" can be used as a conjunctive adverb starting a new sentence (like "alternatively" in this one) or preceded by a semicolon.
If any of the other sentences (which are incomplete) is different in that aspect, that might be a reason to prefer it over the first one.
None of them are really idiomatic, per se. However, the third one seems grammatically incorrect. You shouldn't say:
the chances to reach
Instead, your first example works fine:
the chances of reaching
Your second example also works fine. Truly, you can say either the first or second, but avoid the third because it doesn't work, grammatically...
1 & 2 both seem acceptable, but 2 reads more elegantly, possibly because of the alliteration created with women's chances.
These sentences are complex and have many multi-syllable words - it may help the reader therefore, to go with the option that reads more easily, with more natural breathing.
