My first instinct in this question is use Normal approximation because N is large, and P is exactly between 1 and 0. I used the normal approximation, calculated when $p(X\le 19)$ and got 0.8997. The correct answer.
The confusion arrives at the highlighted part. In my head, wouldn't atleast one would mean $1-p(X=0)=P(X\geq 1)$ ?
The highlighted "at least one credit card" corresponds to the "at least one credit card" on lines 1-2.
And the $X$ you have been using is the number of persons, not number of cards.