document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=254821>Question 254821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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?? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #187133 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=187133\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want 2 of one kind, 2 of another kind, and any card for the 5th.\r<BR>\n" );
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document.write( "your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.\r<BR>\n" );
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document.write( "there are 4 suits in a deck and each suit has 13 cards making a total of 4 * 13 = 52.\r<BR>\n" );
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document.write( "That means you have 4 aces, 4 kings, 4 queens, 4 jacks, 4 tens, etc. down to 4 deuces.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "assume it's a pair of aces.\r<BR>\n" );
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document.write( "you can get them as follows:\r<BR>\n" );
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document.write( "ace of clubs and ace of spades<BR>\n" );
document.write( "ace of clubs and ace of diamonds<BR>\n" );
document.write( "ace of clubs and ace of hearts<BR>\n" );
document.write( "= 3 +<BR>\n" );
document.write( "ace of spades and ace of diamonds<BR>\n" );
document.write( "ace of spades and ace of hearts<BR>\n" );
document.write( "= 5 +<BR>\n" );
document.write( "ace of diamonds and ace of hearts<BR>\n" );
document.write( "= 6\r<BR>\n" );
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document.write( "so for each pair, you can do it in 6 ways.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "some examples would be:\r<BR>\n" );
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document.write( "kings and queens<BR>\n" );
document.write( "kings and jacks<BR>\n" );
document.write( "kings and tens<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "once you have your 2 pairs, the total number of ways you can draw the 5th card is (52-8) * 1 = 44\r<BR>\n" );
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document.write( "total possible combinations of different pairings would therefore be:\r<BR>\n" );
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document.write( "6*6*78*44 = 123552\r<BR>\n" );
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document.write( "that's 78 combinations of different pairs.<BR>\n" );
document.write( "each pair can be derived in 6 ways.<BR>\n" );
document.write( "each combination of pairs can therefore be derived in 6*6 = 36 ways.<BR>\n" );
document.write( "you get 6*6*78 possible combinations of pairs.<BR>\n" );
document.write( "for each of those possible combinations of pairs, there are 44 possible ways that the fifth card can happen.\r<BR>\n" );
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document.write( "to make this easier to understand in an example, let's assume your deck of cards had only kings, queens, and jacks in it.\r<BR>\n" );
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document.write( "let's also assume there are only 3 suits (clubs, spades, hearts).\r<BR>\n" );
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document.write( "that would be a total of 9 cards in the deck with 3 suits.\r<BR>\n" );
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document.write( "you want 2 of one kind, 2 of another kind, and any card for the 5th.\r<BR>\n" );
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document.write( "your hand would basically be 2 pairs plus a fifth card that is not the same as either of the pairs.\r<BR>\n" );
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document.write( "there are 3 suits in this deck and each suit has 3 cards making a total of 3 * 3 = 9 cards in the deck.\r<BR>\n" );
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document.write( "That means you have 3 kings 3 queens and 3 jacks.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "assume it's a pair of kings\r<BR>\n" );
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document.write( "you can get them as follows:\r<BR>\n" );
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document.write( "king of clubs and king of spades<BR>\n" );
document.write( "king of clubs and king of hearts<BR>\n" );
document.write( "king of spades and king of hearts\r<BR>\n" );
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document.write( "so for each pair, you can do it in 3 ways.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "those possible combinations of pairings would be:\r<BR>\n" );
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document.write( "kings and queens<BR>\n" );
document.write( "kings and jacks<BR>\n" );
document.write( "queens and jacks\r<BR>\n" );
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document.write( "once you have your 2 pairs, the total number of ways you can draw the 5th card is (9-6) * 1 = 3\r<BR>\n" );
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document.write( "for kings and queens, the fifth card can be:\r<BR>\n" );
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document.write( "jack of clubs<BR>\n" );
document.write( "jack of spades<BR>\n" );
document.write( "jack of hearts\r<BR>\n" );
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document.write( "total possible number of combinations of pairing would therefore be:\r<BR>\n" );
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document.write( "3*3*3*3 = 81\r<BR>\n" );
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document.write( "you have 3 possible combinations of two pairs each.<BR>\n" );
document.write( "each pair has 3 possible ways it can be achieved.<BR>\n" );
document.write( "each combination of pairs can therefore happen in 3*3 = 9 ways.<BR>\n" );
document.write( "total possible combinations of pairing is therefore 9*3 = 27 possible ways.<BR>\n" );
document.write( "for each of these possible combinations of pairings, the fifth card can be picked in 3 ways.<BR>\n" );
document.write( "that's a total of 27*3 = 81 possible ways.\r<BR>\n" );
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document.write( "here are the posible combinations for just 1 of the 3 possible combinations of pairings.\r<BR>\n" );
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document.write( "the 3 possible combination of pairings are:\r<BR>\n" );
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document.write( "kings and queens<BR>\n" );
document.write( "kings and jacks<BR>\n" );
document.write( "queens and jacks\r<BR>\n" );
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document.write( "this pairing will be for kings and queens.\r<BR>\n" );
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document.write( "let k = king<BR>\n" );
document.write( "let q = queen<BR>\n" );
document.write( "let j = jack\r<BR>\n" );
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document.write( "let 1 = clubs<BR>\n" );
document.write( "let 2 = spades <BR>\n" );
document.write( "let 3 = hearts\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "k1 = king of clubs, k2 = king of spades, k3 = king of hearts<BR>\n" );
document.write( "q1 = queen of clubs, q2 = queen of spades, q3 = queen of hearts<BR>\n" );
document.write( "j1 = jack of clubs, j2 = jack of spades, j3 = jack of hearts\r<BR>\n" );
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document.write( "for kings paired with queens, possible combinations of pairings are:\r<BR>\n" );
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document.write( "k1 k2 q1 q2 j1   k1 k2 q1 q3 j1   k1 k2 q2 q3 j1\r\n" );
document.write( "k1 k2 q1 q2 j2   k1 k2 q1 q3 j2   k1 k2 q2 q3 j2\r\n" );
document.write( "k1 k2 q1 q2 j3   k1 k2 q1 q3 j3   k1 k2 q2 q3 j3\r\n" );
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document.write( "k1 k3 q1 q2 j1   k1 k3 q1 q3 j1   k1 k3 q2 q3 j1\r\n" );
document.write( "k1 k3 q1 q2 j2   k1 k3 q1 q3 j2   k1 k3 q2 q3 j2\r\n" );
document.write( "k1 k3 q1 q2 j3   k1 k3 q1 q3 j3   k1 k3 q2 q3 j3\r\n" );
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document.write( "k2 k3 q1 q2 j1   k2 k3 q1 q3 j1   k2 k3 q2 q3 j1\r\n" );
document.write( "k2 k3 q1 q2 j2   k2 k3 q1 q3 j2   k2 k3 q2 q3 j2\r\n" );
document.write( "k2 k3 q1 q2 j3   k2 k3 q1 q3 j3   k2 k3 q2 q3 j3\r\n" );
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document.write( "</PRE>\r<BR>\n" );
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document.write( "for k1k2 you have 3 possible combinations with queens.<BR>\n" );
document.write( "those would be q1q2, q1q3, q2q3.\r<BR>\n" );
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document.write( "for kik3 you have 3 more possible combinations with queens.<BR>\n" );
document.write( "those would be q1q2, q1q3, q2q3.\r<BR>\n" );
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document.write( "for k2k3 you have 3 more possible combinations with queens.<BR>\n" );
document.write( "those would be q1q2, q1q3, q2q3.\r<BR>\n" );
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document.write( "that's a total of 3*3 = 9 possible combinations with kings and queens.\r<BR>\n" );
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document.write( "for each one of those combinations you have 3 possible combinations of the fifth card.\r<BR>\n" );
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document.write( "it can be j1, j2, or j3.\r<BR>\n" );
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document.write( "that makes a total of 9*3 = 27 possible combinations for kings and queens alone.\r<BR>\n" );
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document.write( "you can see that for kings and queens there are 3*3*3 = 27 possible hands that can be dealt.\r<BR>\n" );
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document.write( "kings can also be paired with jacks which would yield another 27.\r<BR>\n" );
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document.write( "queens can also be paired with jacks which would yield another 27.\r<BR>\n" );
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document.write( "total possible pairings would be 81.\r<BR>\n" );
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document.write( "same principle applies to the larger deck of 4 suits with 13 cards in each suit.\r<BR>\n" );
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document.write( "with the larger deck (52 cards, 13 suits, 4 of the same card in each suit), you get the following:\r<BR>\n" );
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document.write( "you have a total of 6 possible ways to get each pairing.\r<BR>\n" );
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document.write( "each combination of pairing can happen in 6*6 = 36 ways.\r<BR>\n" );
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document.write( "there are a total of 78 possible combinations of pairings.\r<BR>\n" );
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document.write( "that makes a total of 78*36 = 2808 possible combinations of pairings.\r<BR>\n" );
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document.write( "for each of those possible combinations of pairings, you have a choice of 44 different cards to be the fifth card.\r<BR>\n" );
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document.write( "that makes a total of 2808 * 44 = 123552 possible ways you can get a hand with 2 pairs only.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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