document.write( "Question 1184450: 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)? \n" ); document.write( "

Algebra.Com's Answer #849852 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "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,
\n" ); document.write( "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,
\n" ); document.write( "if the 2 sisters are not allowed to be in the same room (say, for academic integrity)?
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\n" ); document.write( "\n" ); document.write( " In the post by @CPhill, the strategy is good, but implementation has errors in calculations.\r
\n" ); document.write( "\n" ); document.write( " I came to fix and to correct his solution. My editing is right in the text.\r
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document.write( "Here's how to solve this problem:\r\n" );
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document.write( "**1. Total Ways to Arrange Students (Without Restriction):**\r\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\n" );
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document.write( "*   Room A: Choose 2 students out of 11:  ¹¹C₂ = (11!)/(2!9!) = 55 ways\r\n" );
document.write( "*   Room B: Choose 3 students from the remaining 9: ⁹C₃ = (9!)/(3!6!) = 84 ways\r\n" );
document.write( "*   Room C: The remaining 6 students go to room C: ⁶C₆ = 1 way\r\n" );
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document.write( "Multiply these together to get the total number of ways without restrictions: 55 * 84 * 1 = 4620\r\n" );
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document.write( "**2. Ways Sisters are Together:**\r\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\n" );
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document.write( "*   **Sisters in Room A:**\r\n" );
document.write( "    *   Choose the 2 sisters for Room A: 1 way\r\n" );
document.write( "    *   Choose the remaining student for Room A: ⁹C₁ = 9 ways  <<<---===  error: there are no more places in A\r\n" );
document.write( "    *   Choose 3 students for Room B: ⁸C₃ = 56 ways            <<<---===  error: should be C(9,3) = 84\r\n" );
document.write( "    *   The remaining 6 students go to Room C.\r\n" );
document.write( "    *   Total ways: 1 * 9 * 56 * 1 = 504 ways                  <<<---===  error: should be 1*84 = 84.\r\n" );
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document.write( "*   **Sisters in Room B:**\r\n" );
document.write( "    *   Choose the 2 sisters for Room B: 1 way\r\n" );
document.write( "    *   Choose the remaining student for Room B: ⁹C₁ = 9 ways   \r\n" );
document.write( "    *   Choose 2 students for Room A: ⁸C₂ = 28 ways\r\n" );
document.write( "    *   The remaining 6 students go to Room C.\r\n" );
document.write( "    *   Total ways: 1 * 9 * 28 * 1 = 252 ways\r\n" );
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document.write( "*   **Sisters in Room C:**\r\n" );
document.write( "    *   Choose the 2 sisters for Room C: 1 way\r\n" );
document.write( "    *   Choose 4 more students for Room C: ⁹C₄ = 126 ways\r\n" );
document.write( "    *   Choose 2 students for Room A: ⁵C₂ = 10 ways\r\n" );
document.write( "    *   The remaining 3 students go to Room B.\r\n" );
document.write( "    *   Total ways: 1 * 126 * 10 * 1 = 1260 ways\r\n" );
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document.write( "Add the ways the sisters can be together in each room: 504 + 252 + 1260 = 2016 ways.  <<<---=== should be 84 + 252 + 1260 = 1596\r\n" );
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document.write( "**3. Ways Sisters are Separated:**\r\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\n" );
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document.write( "4620 - 2016 = 2604                                                                    <<<---=== should be 4620 - 1596 = 3024\r\n" );
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document.write( "Therefore, there are     ways to divide the students such that the sisters are not in the same room.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Fixed and corrected.\r
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