document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022633>Question 1022633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A committee consists of 10 women and 7 men. Three members are chosen as officers. What is the probability that all three officers are women? What is the probability that one of the officers chosen is a woman and the other two are men? What is the probability that at lease one officer is a man? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #638282 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=638282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "A committee consists of 10 women and 7 men. Three members are chosen as officers. <BR>\n" );
document.write( "(a) What is the probability that all three officers are women? <BR>\n" );
document.write( "(b)What is the probability that one of the officers chosen is a woman and the other two are men? <BR>\n" );
document.write( "(c) What is the probability that at lease one officer is a man?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Given 10 women and 7 men, from which 3 are chosen as officers.<BR>\n" );
document.write( "Use C(n,r)=n!/(r!(n-r)!) to represent number of combinations of taking r objects out of n.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a)All 3 are women<BR>\n" );
document.write( "Ways to choose 3 women out of 7 = C(7,3)<BR>\n" );
document.write( "Ways to choose from 17 members = C(17,3)<BR>\n" );
document.write( "Probability of choosing 3 women out of 17 <BR>\n" );
document.write( "= C(10,3)/C(17,3)<BR>\n" );
document.write( "= 120/680<BR>\n" );
document.write( "= 3/17<BR>\n" );
document.write( "=0.17647 (approx.)\r<BR>\n" );
document.write( "\n" );
document.write( "(b) One woman and two men<BR>\n" );
document.write( "Ways to choose 1 woman out of 10 = C(10,1) = 10<BR>\n" );
document.write( "Ways to choose 2 men out of 7 = C(7,2) = 21<BR>\n" );
document.write( "Ways to choose 1 woman and 2 men = 10*21=210<BR>\n" );
document.write( "Probability of choosing 1 woman and 2 men = 210/680 = 21/68 = 0.3088 approx.\r<BR>\n" );
document.write( "\n" );
document.write( "(c) At least one officer is a man.<BR>\n" );
document.write( "This is the complement of case (a), therefore, the <BR>\n" );
document.write( "Probability = 1-3/17 = 14/17 = 0.82353 (approx.) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );