document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177980>Question 1177980</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car is parked among N cars in a row, not at either end. On this return the owner finds that exactly r<BR>\n" );
document.write( "of the N places are still occupied. What is the probability that both neighbouring places are empty. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850391 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=850391\">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( "Understanding the Problem\r<BR>\n" );
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document.write( "Total Cars: N<BR>\n" );
document.write( "Our Car: Parked among N cars, not at either end.<BR>\n" );
document.write( "Remaining Cars: r (including our car)<BR>\n" );
document.write( "Goal: Find the probability that both neighboring places are empty.<BR>\n" );
document.write( "1. Total Possible Arrangements\r<BR>\n" );
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document.write( "We know that r spots are occupied, including our car. Since our car's position is fixed, we need to choose the remaining r-1 spots from the N-1 available spots (excluding our car).\r<BR>\n" );
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document.write( "Total ways to choose r-1 spots from N-1 spots: C(N-1, r-1) = (N-1)! / [(r-1)! * (N-r)!]<BR>\n" );
document.write( "2. Favorable Arrangements\r<BR>\n" );
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document.write( "We want to find the arrangements where our car's two neighboring spots are empty.\r<BR>\n" );
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document.write( "Fix Our Car: Our car's position is fixed.\r<BR>\n" );
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document.write( "Empty Neighbors: The two spots next to our car must be empty.\r<BR>\n" );
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document.write( "Remaining Spots: We have N - 3 remaining spots (excluding our car and its neighbors).\r<BR>\n" );
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document.write( "Remaining Cars: We need to choose r - 1 spots from the N - 3 spots.\r<BR>\n" );
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document.write( "Favorable ways to choose r-1 spots from N-3 spots: C(N-3, r-1) = (N-3)! / [(r-1)! * (N-r-2)!]\r<BR>\n" );
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document.write( "3. Calculate the Probability\r<BR>\n" );
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document.write( "The probability is the ratio of favorable arrangements to total arrangements:\r<BR>\n" );
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document.write( "Probability = C(N-3, r-1) / C(N-1, r-1)<BR>\n" );
document.write( "Probability = [(N-3)! / ((r-1)! * (N-r-2)!)] / [(N-1)! / ((r-1)! * (N-r)!)]<BR>\n" );
document.write( "Probability = [(N-3)! / (N-r-2)!] * [(N-r)! / (N-1)!]<BR>\n" );
document.write( "Probability = [(N-3)! / (N-1)!] * [(N-r)! / (N-r-2)!]<BR>\n" );
document.write( "Probability = [1 / ((N-1)(N-2))] * [(N-r)(N-r-1)]<BR>\n" );
document.write( "Probability = [(N-r)(N-r-1)] / [(N-1)(N-2)]<BR>\n" );
document.write( "Therefore, the probability that both neighboring places are empty is [(N-r)(N-r-1)] / [(N-1)(N-2)]. </SPAN>\n" );
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