"I take out flowers from the coat."
Because から = from. From something, from somewhere - it's always から.
I mean...that's not totally true, though.
「母にプレゼントをもらいます。」, for instance, is fine, and yet the English translation would be "I receive a present from my mother."
So why doesn't に work in this instance?
Because the flowers are physically contained within the coat. The coat isn't giving you flowers, you are taking flowers out of the coat.
What the hell is up with these weird sentences? Flowers from hats and from coats - what's next? I know it's grammar and vocab practice, but since when has that meant completely giving up on making sense?