document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210212>Question 1210212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851625 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=851625\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the positive integer have digits d <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " d <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " …d <BR>\n" );
document.write( "k<BR>\n" );
document.write( "​<BR>\n" );
document.write( "  such that 9≥d <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " >d <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " >⋯>d <BR>\n" );
document.write( "k<BR>\n" );
document.write( "​<BR>\n" );
document.write( " ≥0 and d <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +d <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +⋯+d <BR>\n" );
document.write( "k<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =6.<BR>\n" );
document.write( "The digits are strictly decreasing, so all digits are distinct.<BR>\n" );
document.write( "The sum of the digits is 6.\r<BR>\n" );
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document.write( "We consider the number of even digits in the set of digits whose sum is 6.\r<BR>\n" );
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document.write( "Case 1: Zero even digits (all odd digits)<BR>\n" );
document.write( "The possible distinct odd digits are 1, 3, 5.<BR>\n" );
document.write( "The subsets of {1, 3, 5} whose sum is 6 are:<BR>\n" );
document.write( "\begin{itemize}<BR>\n" );
document.write( "\item Size 1: { } (sum is 0, not 6)<BR>\n" );
document.write( "\item Size 2: {1, 5} (sum is 6). Integer: 51<BR>\n" );
document.write( "\item Size 3: {1, ?, ?} (minimum sum is 1+3+5 = 9 > 6)<BR>\n" );
document.write( "\end{itemize}<BR>\n" );
document.write( "So, from this case, we have the integer 51.\r<BR>\n" );
document.write( "\n" );
document.write( "Case 2: One even digit<BR>\n" );
document.write( "The possible distinct even digits are 0, 2, 4, 6, 8.<BR>\n" );
document.write( "The possible distinct odd digits are 1, 3, 5, 7, 9.\r<BR>\n" );
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document.write( "Subcase 2.1: One even digit<BR>\n" );
document.write( "We need a set of distinct digits including exactly one even digit, whose sum is 6.<BR>\n" );
document.write( "\begin{itemize}<BR>\n" );
document.write( "\item Size 1: {6}. Integer: 6<BR>\n" );
document.write( "\item Size 2: {e,o} where e+o=6. Possible pairs (even, odd) with e>o: (4, 2) - not strictly decreasing, (6, 0) - integer 60. Pairs with o>e: (5, 1) - already counted, (3, 3) - not distinct.<BR>\n" );
document.write( "\item Size 3: {e,o <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " ,o <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " } where e+o <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +o <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =6 and o <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " >o <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " . Possible sets: {4,1,?} (sum too small), {2,3,1}. Integer 321. {0,5,1}. Integer 510.<BR>\n" );
document.write( "\item Size 4: {e,o <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " ,o <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " ,o <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " } where e+o <BR>\n" );
document.write( "1<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +o <BR>\n" );
document.write( "2<BR>\n" );
document.write( "​<BR>\n" );
document.write( " +o <BR>\n" );
document.write( "3<BR>\n" );
document.write( "​<BR>\n" );
document.write( " =6. Possible sets: {4,1,?,?}, {2,3,1,0}. Integer 3210.<BR>\n" );
document.write( "\end{itemize}\r<BR>\n" );
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document.write( "Let's enumerate based on the number of digits:<BR>\n" );
document.write( "1 digit: 6 (one even digit)<BR>\n" );
document.write( "2 digits:\r<BR>\n" );
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document.write( "One even, one odd: 51 (zero even), 42 (one even), 60 (one even) 3 digits:<BR>\n" );
document.write( "One even, two odd: 321 (zero even), 510 (one even) 4 digits:<BR>\n" );
document.write( "One even, three odd: 3210 (one even)<BR>\n" );
document.write( "The integers are:<BR>\n" );
document.write( "From Case 1: 51 (0 even digits)<BR>\n" );
document.write( "From Case 2: 6 (1 even digit), 42 (1 even digit), 60 (1 even digit), 321 (0 even digits), 510 (1 even digit), 3210 (1 even digit)\r<BR>\n" );
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document.write( "The positive integers with strictly decreasing digits and at most one even digit, and sum of digits equal to 6 are: 6, 51, 42, 60, 321, 510, 3210.<BR>\n" );
document.write( "There are 7 such integers.\r<BR>\n" );
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document.write( "Final Answer: The final answer is  <BR>\n" );
document.write( "7<BR>\n" );
document.write( "​<BR>\n" );
document.write( " </SPAN>\n" );
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