Warframe

It has been confirmed that the order is important. You must score kills with primary, secondary, then melee. No mix and match!

I got this solo on Mercury, Terminus Sector, Exterminate Grineer. My loadout was mostly starting equipment: the Braton, MK-1 Kunai, and MK1-Bo. The later you do this with better weapons or mods, the easier it'll be. The key is doing it with enemies tightly grouped together, you don't want to run to another solitary enemy to hit with the Kunai or Bo. Alarms will definitely help keep them together instead of running away for cover, so don't be in a particular hurry to shut them off.

If this isn't working out, an infested invasion along with quicker weapons would do the trick. The infested constantly rush you, so there's no need to chase down enemies. Swords and daggers are your best bet for melee, with dual pistols like the Afuris or strong kunai for secondary.

Geoff certainly has the right idea, however I just wanted to add a few tips to help with this achievement.

I will say that I did not need to have my melee weapon equipped for this - just tapping and dropping an enemy worked for me.

TIPS TO MAKE THIS ACHIEVEMENT EASIER TO OBTAIN:

1.) Play on a low level mission
I think this one is fairly obvious - fight against enemies that will go down fast. Earth and Mercury are basically a no-effort-required playground, so once you have your Warframe to a decent level (20+) go to any mission you prefer, get spotted and corral the enemies into a point where you can drop them with ease.

2.) Do not equip a sentinel or a kubrow
I was having trouble at first, and I soon learned it was because my Helios was wrecking everything before I could even pull the trigger . There are mods to make your companions not attack unless someone attacks you, or only attack if you are attacking, but just play it safe and leave them on the ship.

3.) For your firearms, Speed > Damage
Both your primary and your secondary weapons should be something that is a full-auto kind of deal. I attemtped this with a bow (I got the Dread from the Stalker, so yeah, I was too infatuated with it to switch it out for something else ), and eventually got it, but I would advise something that you can spray and pray with, and that goes for your secondary weapons as well. Having dual pistols with full auto is the best for this, however a strong set of kunai will get the job done too.

The ideal setup would be an AR and dual pistols. If you can not go with dual pistols for whatever reason, then go with a kunai. All have blueprints you can purchase from the market for credits (in-game currecny, not real money).

4.) For your melee weapon, Speed > Damage + Range
Having a leveled dagger or dual dagger is what I would advise. Level them up so you can add some nice mods (increase range, damage, attack speed, and anything else you want).

I went with the Dual Fangs simply because they were easy to build, and they have a decent base attack speed already. With a maxed reach mod I believe you gain 60% range, so do not think you will have to be in an enemies face to drop him.

Avoid your heavy weapons here - things like the gram and the ether reaper. Despite how badass they look, they are noticably slower than other weapons, even with a stance equipped. Stick with something with speed as a base, and then work on the other two attributes as opposed to trying to make a slow weapon faster. With the low level enemies, you won't be needing damage as much as speed anyway.

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Once you have the weapons and Warframe set, head over to a level with garbage mobs, and get yourself spotted. Try to find an area with a nice, short corridor for the enemies to run through. The more of them, the better. Simply wait for enough of them, and then start with your primary. A shot or two should do the trick, so be fast, switch after a couple of bullets fire to your secondary. Be sure to run into the crowd while switch and firing, and after a few rounds another one should drop and you should be close enough to melee another trash mob. And you won't have a sentinel or a kuborw stealing your thunder .

Once this is done you should see an in-game message (that looks like an achievement) appear in game, and I believe a reward of some affinity (but I could be wrong - don't quote me there). The actually achievement did not pop for me for a minute or two after that, so if you get the in game message, but no achievement, just be patient and it should pop before the end of the mission.

Questions, corrections and neg vote justifications below.

For this achievement you need to kill 3 enemies using a primary, secondary, and melee weapon within 5 seconds. to obtain this achievement easily you want to make sure to equip a good primary, something that kills fast. i used a rifle (soma prime) but a shotgun like the dragkoon should work. the important thing is being sure that you have throwing/ninja stars as your secondary weapon ( i used hikou) as well as hand to hand gauntlets/brass knuckles as your melee weapon (i used obex) because you can swap between the two fairly quick. so once you have everything equipped you want to start up a low level mission ( i did lith on earth). You want to wait until you see 3 enemies near each other and when you do you want to kill one with your primary than switch to your secondary kill the second enemy and than quickly kill the third with your melee weapon within 5 seconds and the achievement is yours.