document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200459>Question 1200459</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.	A corporation must appoint a president, chief executive officer (CEO), chief financial officer (CFO) and chief operating officer (COO). It must also appoint a planning committee with six different members.  There are 12 qualified candidates, and officers can also serve on the committee.\r<BR>\n" );
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document.write( "(i)	How many different ways can the officers be appointed?<BR>\n" );
document.write( "(ii)	What is the probability of randomly selecting the officers and getting the youngest candidate a president?<BR>\n" );
document.write( "(iii)	How many different ways can the committee be appointed?<BR>\n" );
document.write( "(iv)	What is the probability of randomly selecting the committee members and getting the six youngest of the qualified candidates?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848082 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><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=848082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's break down the calculations for the corporation's appointments.\r<BR>\n" );
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document.write( "**i) How many different ways can the officers be appointed?**\r<BR>\n" );
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document.write( "* This is a permutation problem since the order of the officers (President, CEO, CFO, COO) matters. <BR>\n" );
document.write( "* We have 12 candidates and need to choose 4 for the officer positions.\r<BR>\n" );
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document.write( "* The formula for permutations is: <BR>\n" );
document.write( "   P(n, r) = n! / (n-r)! <BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "      * n is the total number of items (candidates)<BR>\n" );
document.write( "      * r is the number of items to choose (officers)\r<BR>\n" );
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document.write( "* P(12, 4) = 12! / (12-4)! = 12! / 8! = 12 * 11 * 10 * 9 = 11,880\r<BR>\n" );
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document.write( "* **There are 11,880 different ways to appoint the officers.**\r<BR>\n" );
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document.write( "**ii) What is the probability of randomly selecting the officers and getting the youngest candidate a president?**\r<BR>\n" );
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document.write( "* There is only 1 youngest candidate.<BR>\n" );
document.write( "* The probability of selecting the youngest candidate as president is 1 out of the total number of possible presidents (which is the total number of candidates).\r<BR>\n" );
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document.write( "* Probability = 1 / 12 = 0.0833 (approximately)\r<BR>\n" );
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document.write( "* **The probability of randomly selecting the officers and getting the youngest candidate as president is approximately 0.0833 or 8.33%.**\r<BR>\n" );
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document.write( "**iii) How many different ways can the committee be appointed?**\r<BR>\n" );
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document.write( "* This is a combination problem since the order of the committee members does not matter.<BR>\n" );
document.write( "* We have 12 candidates and need to choose 6 for the committee.\r<BR>\n" );
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document.write( "* The formula for combinations is:<BR>\n" );
document.write( "   C(n, r) = n! / (r! * (n-r)!)<BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "      * n is the total number of items (candidates)<BR>\n" );
document.write( "      * r is the number of items to choose (committee members)\r<BR>\n" );
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document.write( "* C(12, 6) = 12! / (6! * (12-6)!) = 12! / (6! * 6!) = 924\r<BR>\n" );
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document.write( "* **There are 924 different ways to appoint the committee.**\r<BR>\n" );
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document.write( "**iv) What is the probability of randomly selecting the committee members and getting the six youngest of the qualified candidates?**\r<BR>\n" );
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document.write( "* There is only 1 possible combination of the six youngest candidates.\r<BR>\n" );
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document.write( "* Probability = 1 / 924 = 0.001082 (approximately)\r<BR>\n" );
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document.write( "* **The probability of randomly selecting the committee members and getting the six youngest of the qualified candidates is approximately 0.001082 or 0.1082%.**\r<BR>\n" );
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document.write( "Let me know if you'd like to explore any other scenarios or have further questions! <BR>\n" );
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