document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=774680>Question 774680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A student takes a test consisting of 5 questions. For each question, she must choose either A, B or C. \r<BR>\n" );
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document.write( "(a) How many possible ways are there for her to complete the test?<BR>\n" );
document.write( "(b) How many of these possibilities include the answer A at least twice?<BR>\n" );
document.write( "(c) How many possible ways are there to complete the test without ever giving the same<BR>\n" );
document.write( "answer to two consecutive questions? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #472434 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=472434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a)<BR>\n" );
document.write( "For each of the 5 questions, there are 3 ways to answer<BR>\n" );
document.write( "The number of possible ways to do the test are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%2A3%2A3%2A3%2A3+=+3%5E5+ ALIGN=MIDDLE  ALT=\"+3%2A3%2A3%2A3%2A3+=+3%5E5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%5E5+=+243+ ALIGN=MIDDLE  ALT=\"+3%5E5+=+243+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(b)<BR>\n" );
document.write( "Just looking at the A column for each of the 5 questions,<BR>\n" );
document.write( "It will either have a check mark or no check mark.<BR>\n" );
document.write( "The number of possible arrangements for the A<BR>\n" );
document.write( "column are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2A2%2A2%2A2%2A2+=+32+ ALIGN=MIDDLE  ALT=\"+2%2A2%2A2%2A2%2A2+=+32+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I can exclude<BR>\n" );
document.write( "A( 0, 0 , 0, 0, 0 )<BR>\n" );
document.write( "A( 1, 0, 0, 0, 0 )<BR>\n" );
document.write( "A( 0, 1 , 0, 0, 0 )<BR>\n" );
document.write( "A( 0, 0 , 1, 0, 0 )<BR>\n" );
document.write( "A( 0, 0, 0, 1, 0 )<BR>\n" );
document.write( "A( 0, 0 , 0, 0, 1 )<BR>\n" );
document.write( "Excluding these 6 cases, that leaves<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+32+-+6+=+26+ ALIGN=MIDDLE  ALT=\"+32+-+6+=+26+\"   DISABLED_style= \"max-width:90%; height:auto;\" > with A appearing at least twice<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "(c)<BR>\n" );
document.write( "I need to exclude:<BR>\n" );
document.write( "(1) A<BR>\n" );
document.write( "(2) A<BR>\n" );
document.write( "----<BR>\n" );
document.write( "(2) A<BR>\n" );
document.write( "(3) A<BR>\n" );
document.write( "----<BR>\n" );
document.write( "(3) A<BR>\n" );
document.write( "(4) A<BR>\n" );
document.write( "----<BR>\n" );
document.write( "(4) A<BR>\n" );
document.write( "(5) A<BR>\n" );
document.write( "--------<BR>\n" );
document.write( "These 4 cases are excluded as well<BR>\n" );
document.write( "as for B's and C's, so there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4%2A3+=+12+ ALIGN=MIDDLE  ALT=\"+4%2A3+=+12+\"   DISABLED_style= \"max-width:90%; height:auto;\" > exclusions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+243+-+12+=+231+ ALIGN=MIDDLE  ALT=\"+243+-+12+=+231+\"   DISABLED_style= \"max-width:90%; height:auto;\" > allowed\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );