document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176882>Question 1176882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At an annual board meeting company X is deciding whether or not to take the company public. Each board member can vote “yes”, “no”, or “abstain.” If there are n-board members, then how many different ways can the votes be cast? Express your answer as a recursive function, f(n). Do not forget to include any necessary base case(s).<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 #850553 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850553\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's craft a recursive function to model this scenario.\r<BR>\n" );
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document.write( "**Recursive Function**\r<BR>\n" );
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document.write( "Let `f(n)` represent the number of different ways the votes can be cast with `n` board members.\r<BR>\n" );
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document.write( "**Base Case**\r<BR>\n" );
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document.write( "* `f(1) = 3`\r<BR>\n" );
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document.write( "If there's only one board member, they have three choices: \"yes\", \"no\", or \"abstain\".\r<BR>\n" );
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document.write( "**Recursive Step**\r<BR>\n" );
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document.write( "For `n > 1` board members:\r<BR>\n" );
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document.write( "* Consider the first board member. They have 3 options.<BR>\n" );
document.write( "* For each of those options, the remaining `n-1` members can vote in `f(n-1)` ways.\r<BR>\n" );
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document.write( "Therefore, the recursive relation is:\r<BR>\n" );
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document.write( "* `f(n) = 3 * f(n-1)`\r<BR>\n" );
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document.write( "**Complete Recursive Function**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "f(n) = {<BR>\n" );
document.write( "   3,              if n = 1<BR>\n" );
document.write( "   3 * f(n-1),    if n > 1<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "**Explanation**\r<BR>\n" );
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document.write( "This function captures the essence of the voting process. Each member adds a factor of 3 (their voting choices) to the total number of possible outcomes. The recursion breaks down the problem by considering one member at a time.\r<BR>\n" );
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document.write( "**Example**\r<BR>\n" );
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document.write( "Let's calculate the number of ways for a small board with 3 members:\r<BR>\n" );
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document.write( "* `f(1) = 3`<BR>\n" );
document.write( "* `f(2) = 3 * f(1) = 3 * 3 = 9`<BR>\n" );
document.write( "* `f(3) = 3 * f(2) = 3 * 9 = 27`\r<BR>\n" );
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document.write( "So, with 3 board members, there are 27 different ways the votes can be cast. </SPAN>\n" );
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