A person drops an egg onto a carpet, and the egg doesn't break. Then she drops the egg onto a tile floor and it breaks. Which floor most likely had a greater amount of force, and a shorter collision time?

The tile floor.

Explanation:

We can think of this as follows:

By the second Newton's law, we know that:

F = m*a

force is equal to mass times acceleration.

Now, as the egg hits the ground, we can think that there is a force that is applied in a very short period of time.

(similar to the case of a bat hitting a baseball)

Ok, the total energy of the impact will be the same for any surface in which the egg falls.

But suppose that the egg falls in a carpet, in this case, the carpet has a kinda large collision time (the time that the force is applied to the egg), this means that we will have a smaller force, acting for a long period of time. Now if the egg falls on something harder, like the tile floor, the collision time will be a lot shorter, then the force must be a lot stronger to compensate, this force is so large, that breaks the shell of the egg.

the tile floor is then the correct answer.