There are a few schools of thought, and I've seen a few of them in action, and below is my best thinking.
Given the situation mentioned originally - your best best is always center of visible mass. 10' coming at you means center of mass/chest since if you cant see that - your tactical situation is very different. Your next most important thing is shooting, and because you never shoot without ensuring where your muzzle is pointed (I'm a combat vet
and I follow this) you are already aiming therefor unload.
A deputy friend of mine always says when they ask him why he fired 10 rounds his answer will always be "I fired until it stopped firing". Someone mentioned the morgue ballistics thing and sure ... one or two .45's might do the trick so bank that 8-14 of them will do the trick. At 10' your last few rounds might be a few inches from the guy.
Massive unloading however doesn't work so well in real life - either "A" there is more than one assailant or "B" those extra rounds go wide and hit someone or something they are not supposed to. Unless you face these situations often you will be shaking violently ... plan to miss, so aim. I'll hit training in a few.
10' is very very very close, if you let someone get that close in a dangerous situation you deserve to be in the conundrum displayed by 3 pages of posts. If you are in a public place and that's how they got that close your better bet is don't draw, disengage or engage with your fists - keeping your weapon side as far from them as possible (hopefully you have a holster that requires a specific direction of pull). Remember you are in public and scream and holler for help. 10 high school freshmen can beat 1 tough guy. The last thing you should want to do is fire in a crowded space.
At the end of the day the best thing you can do IS train for this. The best (never seen it done though) suggestion for getting the shakes out is training where you have to shoot fast and if you miss you get the cattle prod. Never seen it done but I do think I'd be worried about the prod. LEO train the double tap and that because if you empty on the first guy the second guy has you for free.
My favorite drill starts with the coach talking to the shooter - talking about whatever and then they say "shoot" at which point the shooter draws and fires the two COM rounds. (Center Of Mass if you didn't know COM). While doing this they are backing up and to the side - the longer you can make 10' the better! Every now and then the coach says "fail" which is the code for "the first two rounds didnt stop them" and generally we do the head shot. This is based on the logic that when I started to fire the target was [in this case] 10' away. In the time I fired 2 shots he has closed to within striking distance with his knife/fists ect - I don't expect to miss point blank.
Now lets evaluate two more issues that were brought up. If he has a gun the above drill is still your best option. First, he [or she] is advancing which means they probably don't have the most stable aim*. Second you will be moving reducing you the target - and you should be moving towards cover if at all possible.
Change 2 is what if they are wearing a vest? Well, your initial chest shots are still a good idea. Why you ask? Well for starters you generally won't know they are vested until you shoot them there first and they don't go down - though sometimes they go down anyways (which is reason 1 to shoot COM first no matter what), they just get back up but you now had time to get distance, get cover, aim better and fire successive shots. Second, so you see the vest, whats it rated? Is it real? Only one good way to find out. There are plenty of airsoft look a like body armor sets out there ... stops 6mm plastic ... doesn't stop 9mm copper jacketed lead. You're average thug is aware of this but hopes you don't take the time to find out. Even if Joe Thug is wearing some nice Milsurp plates - has he ever had them x-rayed? Were they thrown out by the military for having cracks? Has he been shot or hit there before ("heyo homes, chek dis thingy out, go ahead, shoot at it you gonna see it works"). You're third reason even if you see the vest is shock value. Your assailant may not expect you to be armed, or expect you will pull the trigger (bad Hollywood). Taking one to the chest plate will defiantly disorient momentarily everyone up to Meth-head. Might be fractions of a second, but that's how firefights are. Same goes for just light and sound - you ever fire a handgun in a small room unexpectedly - it will disorient EVERYONE.
*So what if the guy does have practice moving and shooting, is wearing armor and commonly goes by the moniker "Cop" .... drop your gun or he'll drop you
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No one said it thus far and it seems very worth mentioning. Studies have shown that in most shootings - especially by LEOs - the target was hit in the shoulder, hand & arm. Why? You hit what you look at - and most people confronted with an attack are staring at the 'threat' aka the gun/knife/chainsaw. This is why muscle memory and therefor practice are important - if you instinctively always shot center of mass ... when startled and surprised ... you will.
I've never heard the pelvis thing (in either military or LEO years) but it does sound good. Food for thought though I'll tell you about a friend of mine. While in Iraq he took a 7.62x54R in the pelvis, knocked him down but he said he didn't even know he was hit, let alone feel pain for something like 30 seconds to a minute (at which point he said it was
very painful). The pelvis didn't fracture and the round bounced up and out his stomach. Ghastly wound, but he survived and is happy the sniper had bad aim (he was peeing at the time off a rooftop). So the caution to the pelvis shot may be lack of "pain"? It does however clearly knock down the person in practice.
In case you are wondering the story ends with the USAF dropping a 2,000lb bomb. Proof that in any gunfight your best bet is out gun your opponent - be it with an Infantry platoon, radio and the Air Force or a good working firearm, the skills to use it and practice.
I was reading about an incident where a cop was shot in the face (along with something like 14? times) a while back. Initially he was shot in the face and the assailant began to walk away thinking he was dead - its like that in video games right - he shot the assailant who didn't go down either and both ended up fighting it out. He lived because he had the will to live through multiple gunshots even to the face. Apparently his assailant didn't have as strong a will. (I think there was something on it on THR a while back... looking for it. Another example is on youtube/court tv of a traffic stop).
Moral of that story, SSG Bellavia's (read the book House to House), and countless others is if you are in that situation you will have to be ready to defend yourself. If you trained for it and have the will to live your chances are better no matter where you shoot the person.
(sorry it was long... didnt start out to be)