Which firstborn does the kohen redeem?

Pidyon HaBen is a clear מצוות עשה מן התורה, the torah commands bnei yisroel קדש לי כל בכור פטר – כל רחם בבני ישראל “Sanctify to me every firstborn, the first opening of every womb among the children of Israel”. The kohen is thus the direct personal messenger of השם to accept the redemption from the father of the firstborn, as the torah uses the term לי “to me” – to hashem himself.

The torah says פטר כל רחם, the opening of every womb, making a firstborn delivered via c-section exempt from the pidyon requirement.

Miscarriage

The Mishnah states that a firstborn after a miscarriage of a 40-day old fetus is not required to be redeemed. If the fetus is less than forty days old, the son born afterward needs to be redeemed. This is ruled on in Shulchan Aruch Yoreh De’ah Siman 305, Se’if 23. If the fetus’s age in terms of 40 days can only be approximated the son is exempt from pidyon ha-ben, out of doubt (Se’if 13).

The Rema (23) qualifies that if the limbs of the fetus are still unformed, the son that follows him is obligated for redemption. The Noda Biyhuda (siman 188), writes that until the end of the first trimester, it cannot be assumed that the fetus has formed limbs, making the following firstborn obligated for redemption (yet the father does not make a bracha).

other exemptions;

• A mother who is a bat kohen or a bat levi
• The firstborn of a kohen or levi father
• A firstborn born to a non-Jewish mother