document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190992>Question 1190992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten people (5 men and 5 women) are attending a dinner party. In how many ways can they be arranged if:\r<BR>\n" );
document.write( "\n" );
document.write( "a. they are to be arranged in a line for a picture?\r<BR>\n" );
document.write( "\n" );
document.write( "b. They are to be arranged with men and women alternating around the table?\r<BR>\n" );
document.write( "\n" );
document.write( "c. they are arranged in a line for a picture but the man and women must alternate? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849175 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=849175\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Arrangements in a line:**\r<BR>\n" );
document.write( "\n" );
document.write( "There are 10 people in total, so there are 10! (10 factorial) ways to arrange them in a line.\r<BR>\n" );
document.write( "\n" );
document.write( "10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800\r<BR>\n" );
document.write( "\n" );
document.write( "**b. Arrangements with men and women alternating around a table:**\r<BR>\n" );
document.write( "\n" );
document.write( "Since there are 5 men and 5 women, they can alternate in two ways: M W M W M W M W M W or W M W M W M W M W M. \r<BR>\n" );
document.write( "\n" );
document.write( "First, arrange the men. There are 5 men, so there are 5! ways to arrange them. Similarly, there are 5! ways to arrange the women.\r<BR>\n" );
document.write( "\n" );
document.write( "So, the total number of arrangements is 2 × 5! × 5! = 2 × 120 × 120 = 28,800.\r<BR>\n" );
document.write( "\n" );
document.write( "**c. Arrangements in a line with men and women alternating:**\r<BR>\n" );
document.write( "\n" );
document.write( "Similar to the table arrangement, there are two possible alternating patterns: M W M W M W M W M W or W M W M W M W M W M.\r<BR>\n" );
document.write( "\n" );
document.write( "The number of ways to arrange the men is 5! and the number of ways to arrange the women is 5!.\r<BR>\n" );
document.write( "\n" );
document.write( "So, the total number of arrangements is 2 × 5! × 5! = 2 × 120 × 120 = 28,800.\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Answers:**\r<BR>\n" );
document.write( "\n" );
document.write( "a. 3,628,800 ways<BR>\n" );
document.write( "b. 28,800 ways<BR>\n" );
document.write( "c. 28,800 ways<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );