document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192165>Question 1192165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many ways are there to choose two twos from a standard 52-card deck? <BR>\n" );
document.write( "How many ways are there to choose three cards from a standard 52-card deck without choosing any twos? <BR>\n" );
document.write( "How many five-card hands (drawn from a standard 52-card deck) contain exactly two twos? <BR>\n" );
document.write( "How many five-card hands (drawn from a standard 52-card deck) contain a two-of-a-kind? <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 #849075 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849075\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each part of the problem:\r<BR>\n" );
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document.write( "**1. Choosing Two Twos:**\r<BR>\n" );
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document.write( "* There are four twos in a standard 52-card deck.<BR>\n" );
document.write( "* We want to choose two of them.<BR>\n" );
document.write( "* The number of ways to do this is given by the combination formula \"4 choose 2\":\r<BR>\n" );
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document.write( "   ⁴C₂ = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6\r<BR>\n" );
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document.write( "**2. Choosing Three Cards Without Any Twos:**\r<BR>\n" );
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document.write( "* There are 48 cards in the deck that are *not* twos (52 total cards - 4 twos).<BR>\n" );
document.write( "* We want to choose three of these 48 cards.<BR>\n" );
document.write( "* The number of ways to do this is:\r<BR>\n" );
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document.write( "   ⁴⁸C₃ = 48! / (3! * 45!) = (48 * 47 * 46) / (3 * 2 * 1) = 17296\r<BR>\n" );
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document.write( "**3. Five-Card Hands with Exactly Two Twos:**\r<BR>\n" );
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document.write( "* First, we choose two twos (as calculated in part 1): ⁶C₂ = 6 ways.<BR>\n" );
document.write( "* Then, we need to choose the remaining three cards from the 48 non-two cards (as calculated in part 2): ⁴⁸C₃ = 17296 ways.<BR>\n" );
document.write( "* To get the total number of five-card hands with exactly two twos, we multiply these two results together:\r<BR>\n" );
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document.write( "   6 * 17296 = 103776\r<BR>\n" );
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document.write( "**4. Five-Card Hands with a Two-of-a-Kind:**\r<BR>\n" );
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document.write( "This is trickier because it includes more than just two twos. It means *any* pair (two of the same rank) but not three or four of a kind.\r<BR>\n" );
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document.write( "* **Choose the rank for the pair:** There are 13 ranks in a deck (Ace through King). We choose one of these ranks: ¹³C₁ = 13 ways.<BR>\n" );
document.write( "* **Choose the two cards for the pair:** There are four cards of the chosen rank. We choose two of them: ⁴C₂ = 6 ways.<BR>\n" );
document.write( "* **Choose the ranks for the other three cards:** We need three more cards, each of a different rank, and these ranks must be different from the rank we chose for the pair. There are 12 remaining ranks, and we choose three of them: ¹²C₃ = 220 ways.<BR>\n" );
document.write( "* **Choose the suit for each of the three remaining cards:**  For each of the three cards, we have 4 suit options. This gives us 4 * 4 * 4 = 4³ = 64 possibilities.\r<BR>\n" );
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document.write( "* **Multiply everything together:** 13 * 6 * 220 * 64 = 1098240\r<BR>\n" );
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document.write( "Therefore, there are 1,098,240 five-card hands with exactly one pair (two of a kind).<BR>\n" );
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