document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=260652>Question 260652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A.)  HOW MANY WAYS ARE THERE TO PICK TWO CARDS (W/O REPLACEMENT) SO THAT AT LEAST ONE OF THEM IS AN ACE?\r<BR>\n" );
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document.write( "I TRIED, 52!/(52-4)!4! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #192006 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=192006\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> order is not important so you're dealing with combinations.\r<BR>\n" );
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document.write( "there are 52 cards in the deck.\r<BR>\n" );
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document.write( "there are 4 aces in the deck.\r<BR>\n" );
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document.write( "you are picking 2 cards out of the deck.\r<BR>\n" );
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document.write( "at least 1 of them needs to be an ace.\r<BR>\n" );
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document.write( "assuming exactly 1 of them is an ace, then each ace can be paired with exactly 48 other cards in the deck that are not aces.\r<BR>\n" );
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document.write( "you have 48 cards that can be paired with the ace of spades.<BR>\n" );
document.write( "you have 48 cards that can be paired with the ace of clubs.<BR>\n" );
document.write( "you have 48 cards that can be paired with the ace of hearts.<BR>\n" );
document.write( "you have 48 cards that can be paired with the ace of diamonds.\r<BR>\n" );
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document.write( "that's a total of 192 possible pairs.\r<BR>\n" );
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document.write( "you can also have 2 aces in the pair.\r<BR>\n" );
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document.write( "this can happen in 6 ways.\r<BR>\n" );
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document.write( "ace of spades with ace of clubs<BR>\n" );
document.write( "ace of spades with ace of hearts<BR>\n" );
document.write( "ace of spades with ace of diamonds<BR>\n" );
document.write( "ace of clubs with ace of hearts<BR>\n" );
document.write( "ace of clubs with ace of diamonds<BR>\n" );
document.write( "ace of hearts with ace of diamonds\r<BR>\n" );
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document.write( "add 6 to 192 and you get a total of 198 pairs of cards that include at least 1 ace in them.\r<BR>\n" );
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document.write( "if you look at this in terms of combinations, here's what you have:\r<BR>\n" );
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document.write( "possible combinations of 2 aces from 4 aces is:\r<BR>\n" );
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document.write( "c(4,2) = 4!/(2!*2!) = 6\r<BR>\n" );
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document.write( "possible combinations of 1 ace and 1 other card is:\r<BR>\n" );
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document.write( "c(4,1) * c(48,1) = 4 * 48 = 192\r<BR>\n" );
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document.write( "add these possible combinations up and you get 198 possible combinations of a pair of cards from a deck of 52 cards where at least one of the4 cards is an ace.\r<BR>\n" );
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document.write( "the total possible combinations of a pair of cards from a deck of 52 cards is equal to:\r<BR>\n" );
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document.write( "c(52,2) = 52! / (2!*50!) = 1326.\r<BR>\n" );
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document.write( "the probability that you will draw a pair of cards with at least 1 ace in there is equal to:\r<BR>\n" );
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document.write( "198/1326 = .149321267\r<BR>\n" );
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document.write( "this is equal to the number of pairs that have an ace in them divided by the total number of pairs possible.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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