Apostrophe in grammar is used mainly for <u>contractions</u> of noun/pronoun+verb.
For example:
it+is would be it's
I+am becomes I'm
I+have becomes I've.
Apostrophe is also used to <u>reveal possession of something by someone</u>.
For example:
Amy's bag- revealing that the bag belongs to Amy.
My dad's car- the car belongs to my dad.
My cat's couch- showing that the couch belongs to my cat.
When we use apostrophe is with plural noun, it might get confusing a bit, but not that difficult.
The plural noun that ends with 's' do not need to to again end with s like in examples "monkeys's noise." The correct form of using apostrophe in this is monkeys' noise.
The correct plural form of using apostrophe is C is woman's clothing.
In C, either it will be ladies lingerie or ladies' lingerie.
The examples given in the question have use apostrophe with the plural nouns, but only A is correct. Because the correct usage of apostrophe with plural nouns is when the noun does not end with 's', then we can use +'s. So, option A i.e., children's problem is correct answer.