document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178006>Question 1178006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> using all digits from 0 to 9, allowing for repetition, how many 3 digit numbers are possible that contain at least one 5? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #807177 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=807177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I suppose a math teacher somewhere wrote this problem....<br><BR>\n" );
document.write( "It's sure a good thing that teacher isn't trying to teach English!!<br><BR>\n" );
document.write( "\"using all digits from 0 to 9, allowing for repetition, how many 3 digit numbers are possible that contain at least one 5?\"<br><BR>\n" );
document.write( "(1) The digits (in our familiar base 10 number system) are 0 through 9; there are 10 of them.  There are clearly NO 3-digit numbers that use all of those digits (\"using all digits from 0 to 9\").  If the numbers are to be 3-digit numbers, then that opening phrase is not only unnecessary but contradictory; it does not belong in the statement of the problem.<br><BR>\n" );
document.write( "(2) The question asks for the number of 3-digit numbers that contain AT LEAST one 5.  Therefore, the \"allowing for repetition\" is superfluous.<br><BR>\n" );
document.write( "(3) What's left after we get rid of the unnecessary first two phrases is \"how many 3 digit numbers are possible that contain at least one 5?\"  In that remaining phrase, the \"... are possible that\" is unnecessary.<br><BR>\n" );
document.write( "So the question is really simply \"How many 3-digit numbers contain at least one 5?\"<br><BR>\n" );
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document.write( "Now that we have gotten past the absurd way the problem was stated, we can solve it....<br><BR>\n" );
document.write( "I'll let you find the answer; here is a strategy:<br><BR>\n" );
document.write( "(1) Count how many numbers have a 5 in the hundreds place.<BR>\n" );
document.write( "(2) Count how many numbers have a 5 in the tens place -- but don't count the ones with a 5 in the hundreds place also, because you have already counted those.<BR>\n" );
document.write( "(3) Count how many numbers have a 5 in the ones place -- but don't count the ones that also have a 5 in either the hundreds place or the tens place, because you have already counted those.<br><BR>\n" );
document.write( "Add the numbers you found in (1), (2), and (3).<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );