"Nous sommes de riches enfants."
On a previous example the rich men I got it wrong because I translated it to de riches hommes. The correct answer was given as des hommes riches. On another problem entirely I got it wrong because I didn't place riche in front of the noun.
The only difference between that problem and this one that I can see is that the example was followed by a question mark. Is that a factor?
As for the placement of riche it doesn't seem to conform to B.A.G.S criteria so I don't know why it would be placed in front.
In this example, the adjective 'riches' is placed before the plural noun because it is subjective as opposed to objective (generally, subjective descriptors are placed before the noun, objective descriptors after the noun). And of course there's a matter of context - what is to indicate that these children are rich in the monetary sense. ;-)
Until your death, you are still someone's child. You may also imagine this sentence in a context where you would be a rich child of Mr & Mrs XYZ who left you their fortune...
However, using "children" or "kids", the definite article is not required, since in French "nous sommes des..." is just the plural of "je suis un...".
the singular of the sentence would be: I am A rich child = je suis UN enfant riche.
"des" is the plural of "un" (or une)
in English, "a" (one) has no plural form, hence: we are rich children = nous sommes des enfants riches (or nous sommes de riches enfants, with "des" changed to "de" in front of an adjective)