Continuity mistake: When Satine says, "It's lady's choice.", from a front view, the Argentinian is very close to the bearded man, but from a back shot they are farther away. (00:18:45)
Sacha
Continuity mistake: During the "Like a Virgin" sequence, when the Duke is sitting on a couch, his hair changes between the shots: From loose to perfectly brushed and straight.
Sacha Continuity mistake: During Satine's first number, Diamonds, when she is on the trapeze, her lipstick is a bright red. It changes to a dark pink and then back to red during the song and is finally pink when she has passed out.
Continuity mistake: During the "Rhythm of the Night" dance sequence you see The Petite Princess singing on the bar in a hindu outfit behind the gnarly men. After that you see her in her can can skirt. Then any time you see the gnarly men she's back in the hindu outfit.
Continuity mistake: In the end of "Your Song" when Satine and Christian return to the Elephant Room, Satine puts her right arm around Christian's neck. Shots from Satine's right side reveal her right forearm (you can see her elbow), but shots from her left side show that her sleeve cover the entire forearm, and uncover it when she doesn't put her arm in Christian's neck anymore (touch his shoulder).
Continuity mistake: When Christian and Satine are up on the elephant and she is about to leave, you see him put his hand on the pole. Then it cuts to another shot and you see him put his hand on it again in a different way.
Answer: If Satine runs, the Duke takes control of the Moulin Rouge and shuts the place down, putting everyone she works with onto the streets. If she stays with the Duke, she'll still die, but the Duke won't have any real reason to shut things down. She could run away with Christian and have a few days of freedom and love with him, but she'd be condemning everyone at the Moulin Rouge to destitution, plus the Duke would probably have Christian hunted down and killed. By staying for her final few days, she saves everyone that she loves, at the price of her own happiness (and that of Christian, but at least she'll save his life).
Tailkinker