Question 254821: how many possible 5-card hands can make out of 2 of 1 kinf, 2 of another kind, and 1 0ther card in a regular 52-card deck??
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you want 2 of one kind, 2 of another kind, and any card for the 5th.
your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.
there are 4 suits in a deck and each suit has 13 cards making a total of 4 * 13 = 52.
That means you have 4 aces, 4 kings, 4 queens, 4 jacks, 4 tens, etc. down to 4 deuces.
since there are 4 suits of the same card, and you need to get 2 of them, then your possible combinations are C(4,2) = 4!/2!2! = 6 possible ways for each pair.
assume it's a pair of aces.
you can get them as follows:
ace of clubs and ace of spades
ace of clubs and ace of diamonds
ace of clubs and ace of hearts
= 3 +
ace of spades and ace of diamonds
ace of spades and ace of hearts
= 5 +
ace of diamonds and ace of hearts
= 6
so for each pair, you can do it in 6 ways.
since there are 13 sets of the same card, the total number of combinations of two different pairs you can make from this deck of card is C(13,2) = 13!/(11!*2!) = 13*12/2 = 13*6 = 78.
some examples would be:
kings and queens
kings and jacks
kings and tens
etc.
once you have your 2 pairs, the total number of ways you can draw the 5th card is (52-8) * 1 = 44
total possible combinations of different pairings would therefore be:
6*6*78*44 = 123552
that's 78 combinations of different pairs.
each pair can be derived in 6 ways.
each combination of pairs can therefore be derived in 6*6 = 36 ways.
you get 6*6*78 possible combinations of pairs.
for each of those possible combinations of pairs, there are 44 possible ways that the fifth card can happen.
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to make this easier to understand in an example, let's assume your deck of cards had only kings, queens, and jacks in it.
let's also assume there are only 3 suits (clubs, spades, hearts).
that would be a total of 9 cards in the deck with 3 suits.
you want 2 of one kind, 2 of another kind, and any card for the 5th.
your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.
there are 3 suits in this deck and each suit has 3 cards making a total of 3 * 3 = 9 cards in the deck.
That means you have 3 kings 3 queens and 3 jacks.
since there are 3 of each kind, and you need to get 2 of them to make a pair, then your possible combinations for each pair are C(3,2) = 3!/1!2! = 3.
assume it's a pair of kings
you can get them as follows:
king of clubs and king of spades
king of clubs and king of hearts
king of spades and king of hearts
so for each pair, you can do it in 3 ways.
the total number of combinations of two different pairing you can make from this deck of card is C(3,2) = 3!/(1!*2!) = 3*2*1/1*2*1 = 6/2 = 3.
those possible combinations of pairings would be:
kings and queens
kings and jacks
queens and jacks
once you have your 2 pairs, the total number of ways you can draw the 5th card is (9-6) * 1 = 3
for kings and queens, the fifth card can be:
jack of clubs
jack of spades
jack of hearts
total possible number of combinations of pairing would therefore be:
3*3*3*3 = 81
you have 3 possible combinations of two pairs each.
each pair has 3 possible ways it can be achieved.
each combination of pairs can therefore happen in 3*3 = 9 ways.
total possible combinations of pairing is therefore 9*3 = 27 possible ways.
for each of these possible combinations of pairings, the fifth card can be picked in 3 ways.
that's a total of 27*3 = 81 possible ways.
here are the posible combinations for just 1 of the 3 possible combinations of pairings.
the 3 possible combination of pairings are:
kings and queens
kings and jacks
queens and jacks
this pairing will be for kings and queens.
let k = king
let q = queen
let j = jack
let 1 = clubs
let 2 = spades
let 3 = hearts
you get:
k1 = king of clubs, k2 = king of spades, k3 = king of hearts
q1 = queen of clubs, q2 = queen of spades, q3 = queen of hearts
j1 = jack of clubs, j2 = jack of spades, j3 = jack of hearts
for kings paired with queens, possible combinations of pairings are:
k1 k2 q1 q2 j1 k1 k2 q1 q3 j1 k1 k2 q2 q3 j1
k1 k2 q1 q2 j2 k1 k2 q1 q3 j2 k1 k2 q2 q3 j2
k1 k2 q1 q2 j3 k1 k2 q1 q3 j3 k1 k2 q2 q3 j3
k1 k3 q1 q2 j1 k1 k3 q1 q3 j1 k1 k3 q2 q3 j1
k1 k3 q1 q2 j2 k1 k3 q1 q3 j2 k1 k3 q2 q3 j2
k1 k3 q1 q2 j3 k1 k3 q1 q3 j3 k1 k3 q2 q3 j3
k2 k3 q1 q2 j1 k2 k3 q1 q3 j1 k2 k3 q2 q3 j1
k2 k3 q1 q2 j2 k2 k3 q1 q3 j2 k2 k3 q2 q3 j2
k2 k3 q1 q2 j3 k2 k3 q1 q3 j3 k2 k3 q2 q3 j3
for k1k2 you have 3 possible combinations with queens.
those would be q1q2, q1q3, q2q3.
for kik3 you have 3 more possible combinations with queens.
those would be q1q2, q1q3, q2q3.
for k2k3 you have 3 more possible combinations with queens.
those would be q1q2, q1q3, q2q3.
that's a total of 3*3 = 9 possible combinations with kings and queens.
for each one of those combinations you have 3 possible combinations of the fifth card.
it can be j1, j2, or j3.
that makes a total of 9*3 = 27 possible combinations for kings and queens alone.
you can see that for kings and queens there are 3*3*3 = 27 possible hands that can be dealt.
kings can also be paired with jacks which would yield another 27.
queens can also be paired with jacks which would yield another 27.
total possible pairings would be 81.
same principle applies to the larger deck of 4 suits with 13 cards in each suit.
with the larger deck (52 cards, 13 suits, 4 of the same card in each suit), you get the following:
you have a total of 6 possible ways to get each pairing.
each combination of pairing can happen in 6*6 = 36 ways.
there are a total of 78 possible combinations of pairings.
that makes a total of 78*36 = 2808 possible combinations of pairings.
for each of those possible combinations of pairings, you have a choice of 44 different cards to be the fifth card.
that makes a total of 2808 * 44 = 123552 possible ways you can get a hand with 2 pairs only.
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