This question is really two questions masquerading as one.

First you're asked how many caps would be produced if the price were P₅? As long as P > AVC, which it clearly is at P₅, the profit maximizing firm produces where P = MC. In this instance, this means producing an output of Q₅... simple enough.

The question then goes on to ask, how much will be produced in long-run equilibrium? To answer this we need to recall what long run equilibrium is for perfect competition. We know that in long run equilibrium firms earn zero profit, which means P = ATC = MC (which is where ATC is at the minimum). This means that in long run equilibrium the price will be P₄, so output will have to be Q₄.

1. Mike's would produce Q₅ in the short run and Q₅ in the long run. Q₅ is right for the short run but is too large for the long run since P₅ cannot be the long run equilibrium price.
2. Mike's would produce Q₅ in the short run and Q₄ in the long run.
3. Mike's would produce Q₄ in the short run and Q₄ in the long run. Q₄ is correct for the long run but is too low an output for the short run.
4. Mike's would produce Q₄ in the short run and Q₂ in the long run. At a price of P₅ Q₄ would be too low an output for profit maximization in the short run and Q₂ is only the correct output if the price is P₂ which only covers variable costs so is too low for a long run equilibrium price.
5. Mike's would produce Q₂ in the short run and Q₂ in the long run. Q₂ is only the correct output if the price is P₂ which only covers variable costs so is too low for a long run equilibrium price.