document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184450>Question 1184450</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A certain class has 11 students, 2 of whom are sisters. The class needs to be split across three rooms for a socially distanced exam, where room A can seat 2 students, room B 3 students, and room C 6 students. How many ways can the students be divided across the rooms, if the 2 sisters are not allowed to be in the same room (say, for academic integrity)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849848 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=849848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**1. Total Ways to Arrange Students (Without Restriction):**\r<BR>\n" );
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document.write( "First, let's calculate the total number of ways to arrange the students without considering the restriction about the sisters. We can think of this as choosing students for each room:\r<BR>\n" );
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document.write( "*   Room A: Choose 2 students out of 11:  ¹¹C₂ = (11!)/(2!9!) = 55 ways<BR>\n" );
document.write( "*   Room B: Choose 3 students from the remaining 9: ⁹C₃ = (9!)/(3!6!) = 84 ways<BR>\n" );
document.write( "*   Room C: The remaining 6 students go to room C: ⁶C₆ = 1 way\r<BR>\n" );
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document.write( "Multiply these together to get the total number of ways without restrictions: 55 * 84 * 1 = 4620\r<BR>\n" );
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document.write( "**2. Ways Sisters are Together:**\r<BR>\n" );
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document.write( "Now, let's calculate the number of ways the sisters *are* together and subtract this from the total. There are two scenarios where the sisters are together:\r<BR>\n" );
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document.write( "*   **Sisters in Room A:**<BR>\n" );
document.write( "    *   Choose the 2 sisters for Room A: 1 way<BR>\n" );
document.write( "    *   Choose the remaining student for Room A: ⁹C₁ = 9 ways<BR>\n" );
document.write( "    *   Choose 3 students for Room B: ⁸C₃ = 56 ways<BR>\n" );
document.write( "    *   The remaining 6 students go to Room C.<BR>\n" );
document.write( "    *   Total ways: 1 * 9 * 56 * 1 = 504 ways\r<BR>\n" );
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document.write( "*   **Sisters in Room B:**<BR>\n" );
document.write( "    *   Choose the 2 sisters for Room B: 1 way<BR>\n" );
document.write( "    *   Choose the remaining student for Room B: ⁹C₁ = 9 ways<BR>\n" );
document.write( "    *   Choose 2 students for Room A: ⁸C₂ = 28 ways<BR>\n" );
document.write( "    *   The remaining 6 students go to Room C.<BR>\n" );
document.write( "    *   Total ways: 1 * 9 * 28 * 1 = 252 ways\r<BR>\n" );
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document.write( "*   **Sisters in Room C:**<BR>\n" );
document.write( "    *   Choose the 2 sisters for Room C: 1 way<BR>\n" );
document.write( "    *   Choose 4 more students for Room C: ⁹C₄ = 126 ways<BR>\n" );
document.write( "    *   Choose 2 students for Room A: ⁵C₂ = 10 ways<BR>\n" );
document.write( "    *   The remaining 3 students go to Room B.<BR>\n" );
document.write( "    *   Total ways: 1 * 126 * 10 * 1 = 1260 ways\r<BR>\n" );
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document.write( "Add the ways the sisters can be together in each room: 504 + 252 + 1260 = 2016 ways.\r<BR>\n" );
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document.write( "**3. Ways Sisters are Separated:**\r<BR>\n" );
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document.write( "Subtract the number of ways the sisters are together from the total number of ways to arrange the students:\r<BR>\n" );
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document.write( "4620 - 2016 = 2604\r<BR>\n" );
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document.write( "Therefore, there are 2604 ways to divide the students such that the sisters are not in the same room.<BR>\n" );
document.write( " </SPAN>\n" );
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