No, no he didnt choke on his food.
Here is the answer: Think about the scene, picture it in your mind. He washes his hands before eating, right?
Poison is on his hands first(because of working with the chemicals), then he turns handle on sink to start water, then he washes his hands(now clean of poison) but when he goes to turn off the handle , he touches the handle which still has poison on it! Get it?