document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191612>Question 1191612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> From a standard deck of 52 cards, how many six-hand cards are possible if there are at least three aces and three other cards?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823441 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823441\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If there are exactly 3 aces in the hand, then there are x = 4 ways to pick these three cards. <BR>\n" );
document.write( "This is equivalent to having x = 4 ways to not pick a particular ace.\r<BR>\n" );
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document.write( "We have n = 52-4 = 48 non-ace cards to pick from and r = 3 slots to fill.<BR>\n" );
document.write( "Use the nCr combination formula.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "48 C 3 = (48!)/(3!*(48-3)!)<BR>\n" );
document.write( "48 C 3 = (48!)/(3!*45!)<BR>\n" );
document.write( "48 C 3 = (48*47*46*45!)/(3!*45!)<BR>\n" );
document.write( "48 C 3 = (48*47*46)/(3!)<BR>\n" );
document.write( "48 C 3 = (48*47*46)/(3*2*1)<BR>\n" );
document.write( "48 C 3 = (103776)/(6)<BR>\n" );
document.write( "48 C 3 = 17296<BR>\n" );
document.write( "We have y = 17296 ways to select the three other non-ace cards.\r<BR>\n" );
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document.write( "In total, there are x*y = 4*17296 = 69,184 ways to select 6 cards such that exactly 3 are aces.\r<BR>\n" );
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document.write( "Let P = 69184\r<BR>\n" );
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document.write( "Now consider the six card hand having exactly 4 aces. There's only one way to select all four aces.\r<BR>\n" );
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document.write( "There are 48 C 2 = 1128 ways to pick the other cards.\r<BR>\n" );
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document.write( "That gives 1*1128 = 1128 ways to have a 6-card hand with exactly 4 aces.\r<BR>\n" );
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document.write( "Let Q = 1128\r<BR>\n" );
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document.write( "Add the previous results:<BR>\n" );
document.write( "P+Q = 69184+1128 = 70,312<BR>\n" );
document.write( "This is the number of six-card hands with at least 3 aces.<BR>\n" );
document.write( "In the event of having 4 aces, we can't have \"3 other cards\" as your teacher mentioned because that would mean having a 7-card hand. \r<BR>\n" );
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document.write( "Answer: 70,312<BR>\n" );
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