Where your carry stands
A fight starts with every unit on the closest enemy. Put your carry where enemies reach it last, and check what its range reaches from there.
Checked for patch 18.4. The numbers come from the live Set 18 data and follow every patch.
The idea
Each board is 4 rows of 7 hexes. When a fight starts, the two boards meet front row to front row, so your back row is 4 hexes from their front row and 7 from their back row.
Every unit starts the fight on the closest enemy. If that enemy is within its range, it attacks from where it stands; if not, it walks toward it until it is. Range is counted in hexes, and set data gives every champion one. Most ranged champions this set have range 4: from your back row that reaches their front row and nothing deeper. The longest is 6, short of the 7 hexes between the two back rows: nothing reaches from one back row to the other until someone moves.
So the first hits land on whoever stands closest to the enemy, and a ranged carry wants to be the unit they reach last. That is the back corner. Enemies have to get through or around your front line to reach it, and only 2 or 3 hexes touch a back corner, against 6 for a hex away from the edges: fewer hexes an enemy can hit it from, and your own units can fill the rest.
The corner has a cost. From there your carry reaches only the enemy units on its own side, so the side you choose decides who it hits first. A melee carry plays by other rules: it has to walk to its target, so it stands beside the front line, on the side it should break through, not in the back.
See it live
Two current boards as they meet in a fight, theirs turned around the way the game puts it. Your carry's range is lit on their board. Move it and read what it reaches.
Pebbles stands in a back corner with range 4. Tap an empty hex of your board to move it.
In range before anyone moves: their Amumu.
The closest enemy starts 4 hexes away.
3 hexes touch it, and your units hold 1 of them. Every open one is a hex an enemy can hit it from.
The rule of thumb
A ranged carry goes in a back corner behind your tanks; a melee carry goes beside the front line. Then check what each one reaches from there.
Check yourself
Pebbles has range 4 and stands in your back row. Which of their rows can it hit before anyone moves?
Pick an answer.