Mathematical beds

We have four beds. What are the sleeping possibilities? Is it a factorial?
4*3*2*1 = 24.
Or are there more since there are five of us? 5! = 120. But we have to add the cats in then, since they are as much of a pain in the ass as the children. 7! = 5040.
Or would it be some combination?
x = number of beds
y = number of participants
x! + y! = xy(no sleep)

In any [mathematical] case not a situation conducive to peaceful sleeping.

We can expand the equation to include

z = fever, nightmare, thirst, loneliness
also z[cat] = feline thirst, hunger, loneliness 

and so

x! + y! + z! + z[cat]! = xyz(no fucking sleep)

which goes like this:

It's midnight. Aidyn cries. Gets out of bed, walks to our room, crying louder with every step.
Which wakes Mia, who begins to cry and cries as she walks to our room. As they arrive at the bed, they start to scream and try to pull me out of bed. "Lait! LAIT!!" Calls for milk wrench me from sleep (although I was of course semi-awake after the first whimper). By now Chloé has heard the cacophony and walks dazed into the room. "Je veux aussi du lait," she mutters. At least we know they're learning French.

This is usually a temporary end of the shenanigans. Milk all around (it's bad, but between midnight and 6am, all you want is for them to shut up and go to sleep) and everyone is asleep in our bed. For a while. Then Mia rolls onto Chloé and Aidyn starts combing my hair with his fingers again. Chloé yells "Mia!" and pushes her away. I yell "Aidyn!" and push him away. He refuses to leave me alone so before he sits up and starts whining for milk again, I turn away and let him at my hair, hoping he'll fall back asleep again soon so that I can leave and go to Chloé's room and sleep in peace.

3am the fun starts anew. (I finally fell asleep so am still hanging on the edge of the bed with Aidyn curled up around my head.) The cat jumps onto the night table and paws things off onto the floor. I want to get up and lock him in the living room before this noise wakes up everyone else, but any movement from me could wake up you-know-who. So I try to indicate to the cat that I am going to KILL him if he doesn't fucking STOP THAT as I slip slllooowwwlllyyy from the bed. I grab the cat but as I'm walking to the door I step on a matchbox car, squeal in pain, and send the car banging into the hollow door. So much for keeping Mr. Fussypants asleep. Aidyn, the cat and I go to the kitchen, get more milk, shut the door with the cat inside and go back to bed/sleep.

5am. Scratch-scratch. Bang! Bang! Scratch-scratch. Bang! Bang! The cat trying to get out of the kitchen door. I again slip carefully from the bed to go into the kitchen and feed the fucking cats so that they will shut up and stop...doing anything at all. It works and we sleep again until the alarm goes off at 6-ish. At that point we are ALL awake because Gaetan can't find his alarm clock since the cat knocked it onto the floor under the bed.

Another night over. Thank a deity. Of course, this was a worst-case scenario, which doesn't happen more than...once a week. 

We've since stopped giving milk in the night and it has (generally) had the expected result: no hysterics for milk since there won't be any. But they still come, they still wake each other and us, and we all tend to sleep poorly and be grouchy and tired after this goes on for too many nights.

I have a few friends with multiple children who were allowed to share their parents' bed. My question for them is: How did you do this? Did you have rules about when to come in? Or why? Or only after/before a certain age? Or did you just have shit sleep until everyone was old enough to deal with their sleeping (or more accurately, waking) issues? What is the best equation for nights with small children?