How come the word "chore" is pronounced like it is, when the word "choreography", which has the same initial letter combination, is pronounced differently? Is there a phonetical rule to explain this?
Because of etymology: chore is written in the New Oxford American Dictionary as “(originally dialect and U.S.): variant of obsolete char or chare (see charwoman)”, which the same pronunciation as the modern verb char. It ultimately comes from Middle English cherre (odd job), from Old English cerr, cierr (turn, occasion).
Choreography, like chorus, choir, all ultimately come from the Latin chorus (pronounced k-) and its derivatives, from Greek khoros (dance).
It is a coincidence, because "chore" has different etymologies. In general, "ch" sounds like a "k" in words originating from Greek; "choreo" from "choreography" derives from the Greek "khoros" (band of dancers or singers, dance, dancing ground).
"ch" has usually one of the three pronunciations: tʃ(porch), k(chaos) or ʃ (machine)
Related question: How do I know when a word with ch is pronounced hard or softly?