Olga is right, and the boy is wrong - he isn't about to die.

McCoy is right too, much to his disgust. His attempts to save the mangled, blackened hand end in a mid-humeral amputation, the damage below that too extensive to undo. After a terse conversation with the Captain (Kirk deeply, deeply hates losing any of his crew, something McCoy can sympathize with), McCoy settles into a chair in the recovery bay, waiting for the boy (Alston, the boy's name is Alston. Mark Alston, Engineer second class) to wake up.