The complete picture is a man dangling a carrot from the end of a stick, in front of the donkey he's riding.

I'm afraid you're confusing the "carrot and stick" approach with the "carrot on a stick" approach.

The former is, in fact, a combination of negative and positive reinforcement. Other langugages have corresponding idioms that describe the same process, but use different expressions for the reward (carrot) and the punishment (stick), for example "sweetmeat and whip", which absolutely cannot be interpreted as "carrot on a stick". You will find that all of these idoms translate to "carrot and stick" in English, as any dictionary will tell you.

If you use "carrot and stick" and mean "carrot on a stick", you'll just be confusing the heck out of whoever you happen to talk to.

http://www.webster.com/dictionar...nd%20stick