Answer:
242,784,360 hands.
Step-by-step explanation:
The number of hands that contain at least one hearts card is given by the total possible number of hands subtracted by the number of hands with no hearts card.
The total possible number of hands is:
[tex]N_T=\frac{52!}{(52-5)!}=\frac{52!}{(47)!} =52*51*50*49*48\\N_T=311,875,200[/tex]
Since there are 13 hearts cards in a deck, the number of possible hands with no hearts card is:
[tex]N_{H=0} =\frac{52-13!}{(52-13-5)!}=\frac{39!}{(34)!} =39*38*37*36*35\\N_{H=0}=69,090,840[/tex]
The number of hands with at least one hearts card is:
[tex]N_{H>0} = N_T-N{H=0}\\N_{H>0}=311,875,200-69,090,840\\N_{H>0}=242,784,360[/tex]
242,784,360 hands contain at least one hearts card.
