How many ways are there to shuffle a deck of cards if cards of the same value (but different suits) are indistinguishable from one another? For instance, the jack of hearts and the jack of spades are considered as the same card?
Initially, my intuition was to take $52!$ and subtract all duplicate shuffles (of which I believe there are $52 * 4!$). $52 * 4!$ comes from $4!$ different ways to arrange each card value, and $52$ different cards. Is this correct?