You're correct that it's not binomial. That's for sampling with replacement, this is sampling without replacement (a 'drawing balls from an urn' or 'drawing from a deck of cards' type model, rather than the more common 'tossing a coin'/'rolling a die' model).
This is just a straight hypergeometric probability calculation. (This is discussed in many basic books on probability.)
See Wikipedia on the hypergeometric distribution.
In particular, in the example (involving Texas Hold 'Em), the event 'the card is a club' corresponds to your 'chose a correct item'.
You can also find some more questions on the hypergeometric here on CV with some searches.
In R, the
phyper function (with
lower.tail=FALSE) evaluates these "$\geq$" probabilities (well, it actually evaluates $>$, but you get $\geq$ by subtracting 1 from the argument). See