document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193717>Question 1193717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many ways can 7 boys and 7 girls be seated at a round table if:\r<BR>\n" );
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document.write( "A. No restriction is imposed?\r<BR>\n" );
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document.write( "B. The girls and the boys are to occupy alternate seats?\r<BR>\n" );
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document.write( "C. 5 particular girls must sit together\r<BR>\n" );
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document.write( "D. 5 particular girls must not sit together?\r<BR>\n" );
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document.write( "E. All the girls must sit together? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848605 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848605\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **A. No Restriction**\r<BR>\n" );
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document.write( "* **Arrange the 14 people in a line:** 14! ways<BR>\n" );
document.write( "* **Account for circular arrangements:** Divide by 14 to account for rotations (since any position can be considered the \"start\")<BR>\n" );
document.write( "* **Number of ways:** 14! / 14 = 13! = 6,227,020,800 ways\r<BR>\n" );
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document.write( "**B. Girls and Boys Alternate**\r<BR>\n" );
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document.write( "* **Arrange the boys in a circle:** 6! ways (circular arrangement)<BR>\n" );
document.write( "* **Place the girls in the 7 spaces between the boys:** 7! ways<BR>\n" );
document.write( "* **Number of ways:** 6! * 7! = 50,400 * 5,040 = 254,016,000 ways\r<BR>\n" );
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document.write( "**C. 5 Particular Girls Must Sit Together**\r<BR>\n" );
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document.write( "* **Treat the 5 girls as a single unit:** Now we have 9 entities to arrange (6 boys + 4 units: 4 individual girls and 1 group of 5 girls)<BR>\n" );
document.write( "* **Arrange the 9 entities in a circle:** 8! ways<BR>\n" );
document.write( "* **Arrange the 5 girls within their group:** 5! ways<BR>\n" );
document.write( "* **Number of ways:** 8! * 5! = 40,320 * 120 = 4,838,400 ways\r<BR>\n" );
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document.write( "**D. 5 Particular Girls Must Not Sit Together**\r<BR>\n" );
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document.write( "* **Find the total number of arrangements (from part A):** 13! ways<BR>\n" );
document.write( "* **Find the number of arrangements where the 5 girls sit together (from part C):** 4,838,400 ways<BR>\n" );
document.write( "* **Number of ways where 5 girls do not sit together:** 13! - 4,838,400 = 6,227,020,800 - 4,838,400 = 6,178,636,800 ways\r<BR>\n" );
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document.write( "**E. All the Girls Must Sit Together**\r<BR>\n" );
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document.write( "* **Treat the 7 girls as a single unit:** Now we have 8 entities to arrange (7 boys + 1 group of 7 girls)<BR>\n" );
document.write( "* **Arrange the 8 entities in a circle:** 7! ways<BR>\n" );
document.write( "* **Arrange the 7 girls within their group:** 7! ways<BR>\n" );
document.write( "* **Number of ways:** 7! * 7! = 5,040 * 5,040 = 25,401,600 ways\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any other questions.<BR>\n" );
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