What’s the Least Amount of Cards To Guarantee a Four-of-a-Kind?

Forty.

The number of decks is irrelevant; the answer is the same if one or one-hundred decks are used.

Any card drawn will be an A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, or K, so there are 13 possibilities each time a card is drawn.

The fastest way to draw a four of a kind is if the first four cards all have the same “value.” The slowest way, which provides the solution, is to first draw 13 three of a kind, and then one more card.

Since 13 x 3 + 1 = 40, if 40 cards are drawn it is guaranteed that those forty cards contain at least one four of a kind.

Source: Mathisfun.com