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switched babies are his sister and half brother sanyan. ml is the son of his father and mother mentioned in the story. its just that he hate his son.

okay so the mother was there in the temple during delivery while the mistress was also there already carrying a baby less than month old but the mother doesn't know it. when the mother gave birth to the child it was announced that a boy have been born but the monk says he saw the child being born was a girl. so based on this you can tell that his father exchanged the child and sent the girl child away while lying to his wife that a baby boy was born.
couldn't remember the details as I watched it mid last year

thus on the day his mother died he made some ridiculous scandal about sangmo so that he will be out of family branch thus the heir of duke tag will automatically go to the second son plus he is known to world as the son of legitimate wife not from some mistress.

You have reached ep 28 then how could you not understand it. did you skipped some ep?? it wasn't about gender. the son was from the father and his lover while daughter is from male lead's mother his true sibling not half. His father never loved his mother as he sees her as his enemy's sister. Married for fame and power. He actually loved the lover but the lover family was implicated for some corruption ( sang mo's uncle punished them) thus his father hate him. he lost contact of his lover and then after some years of his his marriage he found his lover now a musician ( aka music parlour type of thing you see in historical dramas). He had son with his lover and wanted to give him the tag of duke and only the the legitimate son gets that tag.