Five-card hands out of 32

Problem

From a deck of $32$ cards, $5$ cards are drawn. Compute the number of hands containing: (a) no ace; (b) exactly two queens; (c) two spades and three hearts; (d) at least three jacks.

Solution

The order does not matter: we use the simple combinations $\binom{n}{k}$. In the $32$-card deck there are $4$ aces, $4$ queens, $4$ jacks, $8$ spades and $8$ hearts.

(a) No ace: we choose $5$ cards from the $28$ non-aces: $\binom{28}{5} = \boxed{98\,280}$

(b) Exactly two queens: $2$ queens from $4$ and the other $3$ cards from the $28$ non-queens: $\binom{4}{2}\binom{28}{3} = 6\cdot 3276 = \boxed{19\,656}$

(c) Two spades and three hearts: $2$ spades from $8$ and $3$ hearts from $8$: $\binom{8}{2}\binom{8}{3} = 28\cdot 56 = \boxed{1568}$

(d) At least three jacks: the cases «exactly $3$ jacks» and «all $4$ jacks», added together: $\binom{4}{3}\binom{28}{2} + \binom{4}{4}\binom{28}{1} = 4\cdot 378 + 28 = \boxed{1540}$

Links

Topics: Probabilita
Concepts: Combinazioni semplici
Methods: Combinazioni semplici
Skills: Calcolo combinatorio
Exercise type: Problema combinatorio