Question

the children are running _ each other(with,after)

Answer 1

after each other doesn't make any sense
So the answer will be WITH each other.

Answer 2

actually both of the options are right;
The children are running AFTER each other.
The children are running WITH each other.

if after is added to the sentence, it gives a meaning that children are behind each other.
whereas if with is added then it signifies that all of them are running together.

but according to me, AFTER is more appropriate.