The printer’s manuscript has the singular “their chance for learning”, whereas the 1830 edition has the plural “their chances for learning”. Elsewhere in the text, there is no occurrence of the plural chances. But there are six occurrences of the singular chance:
The last example is somewhat similar to the case here in 3 Nephi 6:12 (both are followed by a modifying prepositional phrase headed by for). Still, these six examples are each preceded by either the indefinite article a /an or the negative determiner no, so none of these examples are precisely like the case here in 3 Nephi 6:12 where chance(s) is preceded by their. The plural chances is what we expect in modern English, but the singular does work. Probably what happened here in the transmission of the text is that the 1830 compositor was influenced by the preceding reference to plural individuals in this sentence (“and the people began to be distinguished by ranks according to their riches and their …”); perhaps the compositor was also influenced by the preceding conjoined plural riches to set chances rather than chance. It seems less likely that Oliver Cowdery would have replaced the expected chances with the unexpected chance. The critical text will accept the unexpected but possible singular as the more probable reading here in 3 Nephi 6:12.
Summary: In accord with 𝓟, restore the singular chance in 3 Nephi 6:12; in this sentence, preceding plurals seem to have influenced the 1830 compositor to replace an original chance with chances; the plural reading is what modern English readers expect in this passage.