[MortalWombatDotCom]

At least two different questions have been addressed so far.

1) what is the probability that a flopped set improves to a full house or better by the river.

2) what is the probability that a flopped set improves to a full house or better by the river given my opponent has flopped a straight.

these are not the same question, and their solutions are different.

that having been said, i can't figure out why

1 - (38/45 * (6/32 * 35/44 + 26/32 * 34/44))

would be correct for version 2... i get

1 - (38/45 * (6/38 * 35/44 + 32/38 * 34/44))

to miss your FH or better, you need to miss on the turn (which happens 38 ways out of 45 available cards) and then miss again on the river... of the 38 ways that you might have missed the first time, 6 leave you with 9 outs, the remaining 32 leave you with 10 outs.