document.write( "Question 1147363: How many ways are there to choose five cards from
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Algebra.Com's Answer #769016 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Did you mean EXACTLY two cards with the same number or\r\n" );
document.write( "did you mean AT LEAST two cards with the same number?\r\n" );
document.write( "We cannot tell.\r\n" );
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document.write( "If you meant EXACTLY two cards with the same number:\r\n" );
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document.write( "Choose the single number to have exactly two cards of in 13 ways.\r\n" );
document.write( "Choose the suits for those two cards in 4C2 = 6 ways\r\n" );
document.write( "Choose three of the remaining numbers for the other 3 cards \r\n" );
document.write( "in 12C3=220 ways\r\n" );
document.write( "Choose the suit for the smallest numbered card of those three in 4 ways.\r\n" );
document.write( "Choose the suit for the middle numbered card of those three in 4 ways.\r\n" );
document.write( "Choose the suit for the largest numbered card of those three in 4 ways.\r\n" );
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document.write( "That's 13∙6∙220∙4∙4∙4 = 1098240 ways.\r\n" );
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document.write( "If you meant AT LEAST two cards with the same number:\r\n" );
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document.write( "We first calculate the number of ways for the complement event,\r\n" );
document.write( "the number of ways where there are no two cards with the same number:\r\n" );
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document.write( "Choose 5 different numbers from the 13 in 13C5 = 1287 ways.\r\n" );
document.write( "Choose the suit for the smallest numbered card in 4 ways.\r\n" );
document.write( "Choose the suit for the next to smallest numbered card in 4 ways.\r\n" );
document.write( "Choose the suit for the middle numbered card in 4 ways.\r\n" );
document.write( "Choose the suit for the next to largest numbered card in 4 ways.\r\n" );
document.write( "Choose the suit for the largest numbered card in 4 ways.\r\n" );
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document.write( "That's 1287∙4∙4∙4∙4∙4 = 1317888 ways for the complement event.\r\n" );
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document.write( "We can choose any 5 cards in 52C5 = 2598960 ways.  We subtract the\r\n" );
document.write( "number of ways the complement event can occur:\r\n" );
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document.write( "2598960 - 1317888 = 1281072 ways.\r\n" );
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document.write( "Edwin
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