Name of a multiarmed bandit with only some levers available

In order to model a card game, as an exercise, I was thinking of an elementary setting as a multiarmed bandit, each lever being the distribution of expected rewards of a specific card.

But, of course, the player only has some cards in the hand each round, or, equivalently, for a given round, it has available a number $$n$$ of arms randomly selected from the total number $$N$$ of levers.

Is this just a "contextual bandit" or has it some specific, narrower, name that I could use to look up in the literature?

• To clarify, is there only one state of the game? Can you only select 1 card? – Jaden Travnik Mar 3 '18 at 22:54
• Yep, habitual game mechanics: you have received $n$ random cards from an stack of $N$, you can play one. In some cases, (eg bridge) perhaps even less than $n$. Seems contextual, but very particular. – arivero Mar 6 '18 at 12:30