How many different four-card hands that contain at most two hearts can be dealt from a standard deck of playing cards?

Question

I am in grade 11 IB math and the unit is counting methods

Answer 1

Hi Tan,

 

For "at least" and "at most" combination problems - it's easier to start with the total possible combinations and then subtract the cases we don't want:

    1.  Total Possible Combinations:

             ₅₂C₄ (52 cards taken 4 at a time)

    2.  Three Hearts (unwanted case to subtract):

             ₁₃C₃ x ₃₉C₁ (13 hearts taken 3 at a time multiplied by 39 non-hearts taken 1 at a time)

    3.  Four Hearts (unwanted case to subtract):

             ₁₃C₄ (13 hearts taken 4 at a time)

 

Putting it all together, we have our counting formula for this problem, which we can use to calculate our answer:

    Total Combos - (3-heart combos + 4-heart combos)

    ₅₂C₄ - ((₁₃C₃ x ₃₉C₁) + ₁₃C₄)

    270,725 -((286 x 39) + 715)

    270,725 - (11,154 + 715)

    270,725 - 11,869

    258,856 4-card hands with at most 2 hearts

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan (Romans 5:8)