document.write( "Question 1057124: Suppose you are dealt six cards from a standard deck of 52 playing
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Algebra.Com's Answer #672302 by Edwin McCravy(20059) $\"\"$ $\"About$

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Suppose you are dealt six cards from a standard deck of 52 playing
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document.write( "First we calculate the number of successful hands\r\n" );
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document.write( "Case 1:  The 5S is not included.  \r\n" );
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document.write( "Choose the ranks for the three spades in 12C3 = 220 ways.\r\n" );
document.write( "Choose the suits for the three fives in 3C3 = 1 way \r\n" );
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document.write( "That's 220*1 = 220 ways for case 1\r\n" );
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document.write( "Case 2: The 5 of spades is included.\r\n" );
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document.write( "We choose the 5 of spades 1C1 = 1 way.\r\n" );
document.write( "We choose the ranks for the other 2 spades in 12C2 = 66 ways.\r\n" );
document.write( "We choose the suits for the other 2 fives in 3C2 = 3 ways\r\n" );
document.write( "We choose the one card that is neither a spade nor a five \r\n" );
document.write( "  in 36C1 = 36 ways.\r\n" );
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document.write( "[It's 36 because 52 cards - 13 spades - 3 non-spade 5's = 36]\r\n" );
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document.write( "That's 1*66*6*36 = 14256 for case 2\r\n" );
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document.write( "Total successful hands = 220 + 14256 = 14476\r\n" );
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document.write( "Total possible hands = 52C6 = 20358520   \r\n" );
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document.write( "Probability = 14476/20358520 which reduces to 11/15470.\r\n" );
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document.write( "What is then the probability of being dealt at least 3 spades \r\n" );
document.write( "and 3 fives?\r\n" );
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document.write( "There is only two additional case besides the preceding, \r\n" );
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document.write( "1. the case where all 4 5's are chosen, including the 5 of spades,\r\n" );
document.write( "and 2 more spades.\r\n" );
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document.write( "Choose the 4 5's 4C4 = 1 way\r\n" );
document.write( "Choose the ranks for the other 2 spades 12C2 = 66 ways\r\n" );
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document.write( "That's 1*66 = 66 ways\r\n" );
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document.write( "2. the case where 4 spades are chosen, including the 5 of spades,\r\n" );
document.write( "and 2 more 5's. \r\n" );
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document.write( "Choose the 5 of spades 1 way,\r\n" );
document.write( "Choose the ranks for the other 3 spades 12C3 = 220 ways\r\n" );
document.write( "Choose the other 2 5's 3C2 = 3 ways.\r\n" );
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document.write( "That's 1*220*3 = 660 ways.\r\n" );
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document.write( "That's an additional 66+660 = 726 ways to add to the 14476\r\n" );
document.write( "from the preceding problem, or 14476+726 = 15202 ways.\r\n" );
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document.write( "Total possible hands = 52C6 = 20358520   \r\n" );
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document.write( "Probability = 15202/20358520 which reduces to 7601/10179260. \r\n" );
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document.write( "Edwin

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