document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1069677>Question 1069677</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A typical combination lock is opened with the correct sequence of three numbers between o and 39 inclusive.<BR>\n" );
document.write( "(I) how many different \"combinations\" are possible if a number can be used more than once (like 27-27-27)? *both permutations and combinations require \"without replacement\" so this isn't either of those.<BR>\n" );
document.write( "(II) How many different \"combinations\" are possible if no number can be repeated? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #684848 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=684848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A fifth-grader approach works:<BR>\n" );
document.write( "(I) how many different \"combinations\" are possible if a number can be used more than once (like 27-27-27)?<BR>\n" );
document.write( "There are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39 ALIGN=MIDDLE  ALT=\"39\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the first number,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39 ALIGN=MIDDLE  ALT=\"39\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the second number, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39 ALIGN=MIDDLE  ALT=\"39\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the third number.<BR>\n" );
document.write( "So the total number of possible sequence choices is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2A39%2A39=highlight%2859319%29 ALIGN=MIDDLE  ALT=\"39%2A39%2A39=highlight%2859319%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Maybe your instructor would call that \"permutations with replacement.\"<BR>\n" );
document.write( " <BR>\n" );
document.write( "(II) How many different \"combinations\" are possible if no number can be repeated?<BR>\n" );
document.write( "There are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39 ALIGN=MIDDLE  ALT=\"39\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the first number,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=38 ALIGN=MIDDLE  ALT=\"38\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the second number, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37 ALIGN=MIDDLE  ALT=\"37\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible choices for the third number.<BR>\n" );
document.write( "So the total number of possible sequences without repeated numbers is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2A38%2A37=highlight%2854834%29 ALIGN=MIDDLE  ALT=\"39%2A38%2A37=highlight%2854834%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Maybe your instructor would call that \"permutations without replacement.\"<BR>\n" );
document.write( " <BR>\n" );
document.write( "Combinations, permutations, replacement are words you are required/expected to use in your class.<BR>\n" );
document.write( "You are instructed to identify the \"type of problem\" and apply the solving procedure prescribed for that \"type of problem.\"<BR>\n" );
document.write( "Once your head is filled with someone else's words and problem-solving recipes,<BR>\n" );
document.write( "you try to faithfully execute the programs installed in your head.<BR>\n" );
document.write( "You search for the imposed words and recipes instead of just thinking about the situation as a human would. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );