It is called the Tueller Drill and the distance is 21 feet. Within 21 feet, a knife-wielding attacker stands a very good chance of stabbing the defender due to the time it takes the attacker to cover 21 feet vs. the defender unholstering, aiming and firing their weapon.
Lets try a metal experiment, you have your knife and 3 enemies attack you at the same time. How many can you neutralize from 5 feet with your knife ? Suppose you have special training and you can throw your knife and kill one of them, you still have 2 ... Try the same mental experiment when you have a firearm in your hand ...
You can inflict much more damage with a firearm even without special training (not everybody can throw a knife and do some damages).
That is a poorly specified, unrealistic thought experiment with lots of hidden variables.
Everything hinges on who attacks first. The weapon is secondary.
Knives are extremely dangerous and more difficult for your assailant to control at close range, and it doesn't seem like you have an appreciation of that.
If you have a gun and if you decide to attack first, maybe you can shoot two or three of them, disabling them, before they can kill you. I've never been in combat, but I don't think that kind of scenario is likely to happen in the real world. Also, even fatal shots are not immediately disabling unless they're to the CNS. Slashes across the arms can readily disable someone in a way that shooting them, even in the chest, won't, unless you hit their spine or some critical nerve or muscle area.
Guns are vastly superior if you have the advantage of distance. Otherwise, not so much.
The context of the thread was police officers being trained to be wary of knives. Imagine two comparable traffic stops where in one case the "perp" has a concealed knife, and the other has a concealed gun. A single officer in each case faces a very comparable risk at close range. I'd argue that at three feet, the officer is much more likely to be killed by a concealed knife than a concealed gun, assuming that is the perp's intention (and the perp has already been asked to get out of the car).
5 feet may not be enough for you to shoot more than one person even if you have your weapon already drawn. When fighting opponents at close range you'd probably be more successful with your knife - especially if you know how to handle one. It's easier to hit with a knife than with a bullet in close quarters, and as 'harshreality suggests, you can disable your opponents in ways the gun wouldn't let you, by cutting through their nerves.
Irrelevant. Let's try a mental experiment. Let's say a cop is performing a traffic stop and asks the driver to step out of the car. If the driver steps out and jams a 6 inch knife into the cop's neck, how much does it matter that the cop could theoretically shoot 3 people?
If you are being attacked by three people, you probably shouldn't start by throwing a knife at one of them. You will probably then have two people, one of them armed with your knife.
While this may have an influence, it's important to remember that police officers are also trained to be wary of knives:
https://www.youtube.com/watch?v=J_KJ1R2PCMM
You can't totally disarm a population, so there are diminishing returns on this front.