document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210198>Question 1210198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.\r<BR>\n" );
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document.write( "You are dealt a hand of 13 cards.  Find the probability that your hand has a void.  (Your hand has a void if it does not contain any cards of a particular suit.)\r<BR>\n" );
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document.write( "Once you've computed the answer in terms of binomial coefficients, use a calculator or computer to determine the answer to the nearest tenth of a percent, and enter that as your answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851548 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=851548\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Total Number of Possible Hands**\r<BR>\n" );
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document.write( "* The total number of 13-card hands that can be dealt from a standard 52-card deck is:<BR>\n" );
document.write( "    * ⁵²C₁₃ = 52! / (13! * 39!) ≈ 6.35 × 10¹¹\r<BR>\n" );
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document.write( "**2. Number of Hands with a Void in a Specific Suit**\r<BR>\n" );
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document.write( "* Let's say we want to find the number of hands with a void in spades.<BR>\n" );
document.write( "* If there are no spades, we are choosing 13 cards from the remaining 39 cards (52 - 13).<BR>\n" );
document.write( "    * ³⁹C₁₃ = 39! / (13! * 26!) ≈ 8.12 × 10¹⁰\r<BR>\n" );
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document.write( "**3. Number of Hands with a Void in Any Suit**\r<BR>\n" );
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document.write( "* Since there are 4 suits, we might think the total number of hands with a void is 4 * ³⁹C₁₃. However, we need to account for overlaps (hands with voids in multiple suits).\r<BR>\n" );
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document.write( "**4. Inclusion-Exclusion Principle**\r<BR>\n" );
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document.write( "We'll use the inclusion-exclusion principle to account for overlaps.\r<BR>\n" );
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document.write( "* Let A, B, C, D represent the events of having a void in spades, hearts, diamonds, and clubs, respectively.<BR>\n" );
document.write( "* |A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D| - |A ∩ B| - ... + |A ∩ B ∩ C| - |A ∩ B ∩ C ∩ D|\r<BR>\n" );
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document.write( "* **|A| = |B| = |C| = |D| = ³⁹C₁₃** (Number of hands with a void in one suit)<BR>\n" );
document.write( "* **|A ∩ B| = |A ∩ C| = ... = ²⁶C₁₃** (Number of hands with voids in two suits)<BR>\n" );
document.write( "* **|A ∩ B ∩ C| = |A ∩ B ∩ D| = ... = ¹³C₁₃** (Number of hands with voids in three suits)<BR>\n" );
document.write( "* **|A ∩ B ∩ C ∩ D| = 0** (It's impossible to have voids in all four suits with a 13-card hand)\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "* |A ∪ B ∪ C ∪ D| = 4 * ³⁹C₁₃ - ⁶C₂ * ²⁶C₁₃ + ⁴C₃ * ¹³C₁₃\r<BR>\n" );
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document.write( "**5. Calculate the Values**\r<BR>\n" );
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document.write( "* ³⁹C₁₃ = 8122425444<BR>\n" );
document.write( "* ²⁶C₁₃ = 10400600<BR>\n" );
document.write( "* ¹³C₁₃ = 1\r<BR>\n" );
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document.write( "* |A ∪ B ∪ C ∪ D| = 4 * 8122425444 - 6 * 10400600 + 4 * 1<BR>\n" );
document.write( "* |A ∪ B ∪ C ∪ D| = 32489701776 - 62403600 + 4<BR>\n" );
document.write( "* |A ∪ B ∪ C ∪ D| = 32427298180\r<BR>\n" );
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document.write( "**6. Calculate the Probability**\r<BR>\n" );
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document.write( "* Probability = |A ∪ B ∪ C ∪ D| / ⁵²C₁₃<BR>\n" );
document.write( "* Probability = 32427298180 / 635013559600 ≈ 0.051065\r<BR>\n" );
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document.write( "**7. Convert to Percentage and Round**\r<BR>\n" );
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document.write( "* 0.051065 * 100% ≈ 5.1065%<BR>\n" );
document.write( "* Rounded to the nearest tenth of a percent: 5.1%\r<BR>\n" );
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document.write( "**Therefore, the probability that your hand has a void is approximately 5.1%.**<BR>\n" );
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