[Math] Combination: Card Problems

combinatoricspermutations

1) How many eight-card hands can be chosen from exactly 2 suits of an ordinary 52-card deck? (there are 4 suits clubs, diamonds,
hearts and spades

I think since there are 26 cards in 2 suits and eight cards from those 26 (order does not matter), thus
$C(26,8)/C(52,26)$ ?

2) How many 13-card bridge hands can be chosen from an ordinary
52-card deck that contain six cards of one suit and four and three
cards of another two suits? (there are 4 suits clubs, diamonds,
hearts and spades

I do not understand 2nd problem.

Best Answer

  1. I don't know why you are dividing by $\binom{52}{26}$. The number $\binom{26}{8}$ already counts the number of 8-card hands you could get from a pre-specified $26$ cards. You may need to multiply by $\binom{4}{2}$ to choose which two suits to use, though.

  2. There are $4 \cdot 3 \cdot 2$ ways to choose the 6-card suit, the 4-card suit, and the 3-card suit to make up your hand. Then you multiply by $\binom{13}{6}$, $\binom{13}{4}$, and $\binom{13}{3}$ to choose the number of cards of each suit.

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