document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=634813>Question 634813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many three digit even numbers less than 500 can be formed from the integers 1, 2, 3, 4, 5, 6, and 7 if the integers cannot be used more than once? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #399976 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=399976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are four scenarios:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Scenario A)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You choose 2 even numbers for the hundreds and tens digits. So you'll have one even number left over (since there are only 3 even numbers in this list). So there are 3 ways to play out this scenario\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since something like 246 is different from 426, this means that we double the count to 2*3 = 6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "BUT\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we don't count something like 642 or 624 since we can't go over 500.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the count so far is 6 - 2 = 4\r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Scenario B)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You chose one even number for the hundreds digit. Then one odd number for the tens digit. Finally you choose one even number for the units digit.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So you have 2 choices (2 or 4) for the hundreds digit.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then you now have 4 choices (1,3,5,7) for the tens digit\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Finally, you have 2 choices for the units digit (2, 4, 6, but kick out either 2 or 4 -- the number you chose for the hundreds digit)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So add on 2*4*2 = 16\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Scenario C)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You choose an odd number for the first digit and an even number\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So you have 2 choices for the hundreds digit (1, 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then you have 3 choices for the tens digit (2, 4, 6)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Finally, you have 2 choices for the units digit (2, 4, 6, but eliminate what you chose for the tens digit)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So add on 2*3*2 = 12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Scenario D)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The hundreds and tens digit are both odd.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So you have 2 choices for the hundreds digit (1 or 3), then you have 3 choices for the tens digit (1, 3, 5, 7...but take off one number), finally you have 3 even digits to choose from\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So you add on 2*3*3 = 18 numbers\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the total number of ways to do this is\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4+16+12+18 = <font color=\"red\">50</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, there are <font color=\"red\">50</font> ways to make an even 3 digit number that is less than 500 using numbers from this list {1, 2, 3, 4, 5, 6, 7} where each number is only used once.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here is the list of the <font color=\"red\">50</font> numbers:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "124<BR>\n" );
document.write( "126<BR>\n" );
document.write( "132<BR>\n" );
document.write( "134<BR>\n" );
document.write( "136<BR>\n" );
document.write( "142<BR>\n" );
document.write( "146<BR>\n" );
document.write( "152<BR>\n" );
document.write( "154<BR>\n" );
document.write( "156<BR>\n" );
document.write( "162<BR>\n" );
document.write( "164<BR>\n" );
document.write( "172<BR>\n" );
document.write( "174<BR>\n" );
document.write( "176<BR>\n" );
document.write( "214<BR>\n" );
document.write( "216<BR>\n" );
document.write( "234<BR>\n" );
document.write( "236<BR>\n" );
document.write( "246<BR>\n" );
document.write( "254<BR>\n" );
document.write( "256<BR>\n" );
document.write( "264<BR>\n" );
document.write( "274<BR>\n" );
document.write( "276<BR>\n" );
document.write( "312<BR>\n" );
document.write( "314<BR>\n" );
document.write( "316<BR>\n" );
document.write( "324<BR>\n" );
document.write( "326<BR>\n" );
document.write( "342<BR>\n" );
document.write( "346<BR>\n" );
document.write( "352<BR>\n" );
document.write( "354<BR>\n" );
document.write( "356<BR>\n" );
document.write( "362<BR>\n" );
document.write( "364<BR>\n" );
document.write( "372<BR>\n" );
document.write( "374<BR>\n" );
document.write( "376<BR>\n" );
document.write( "412<BR>\n" );
document.write( "416<BR>\n" );
document.write( "426<BR>\n" );
document.write( "432<BR>\n" );
document.write( "436<BR>\n" );
document.write( "452<BR>\n" );
document.write( "456<BR>\n" );
document.write( "462<BR>\n" );
document.write( "472<BR>\n" );
document.write( "476\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Again, there are <font color=\"red\">50</font> numbers in that list above<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );