document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210210>Question 1210210</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many positive integers are there whose digits strictly decrease from left to right, 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 #851621 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=851621\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the positive integer have digits $d_1 d_2 \dots d_k$ such that $9 \ge d_1 > d_2 > \dots > d_k \ge 0$ and $d_1 + d_2 + \dots + d_k = 6$.\r<BR>\n" );
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document.write( "Since the digits are strictly decreasing and non-negative, all the digits must be distinct.<BR>\n" );
document.write( "Also, since the sum of the digits is 6, the number of digits cannot be too large. The maximum possible number of digits occurs when the digits are $5, 1$, or $4, 2$, or $3, 2, 1$, or $6$.\r<BR>\n" );
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document.write( "We consider the possible sets of distinct digits whose sum is 6:\r<BR>\n" );
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document.write( "Case 1: One digit<BR>\n" );
document.write( "The only possibility is the digit 6. The number is 6. (1 integer)\r<BR>\n" );
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document.write( "Case 2: Two digits<BR>\n" );
document.write( "We need two distinct digits $d_1 > d_2$ such that $d_1 + d_2 = 6$. The possible pairs are (5, 1) and (4, 2). The corresponding integers are 51 and 42. (2 integers)\r<BR>\n" );
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document.write( "Case 3: Three digits<BR>\n" );
document.write( "We need three distinct digits $d_1 > d_2 > d_3$ such that $d_1 + d_2 + d_3 = 6$. The only possible set of digits is {3, 2, 1}. The corresponding integer is 321. (1 integer)\r<BR>\n" );
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document.write( "Case 4: Four digits<BR>\n" );
document.write( "We need four distinct digits $d_1 > d_2 > d_3 > d_4$ such that $d_1 + d_2 + d_3 + d_4 = 6$. The only possible set of digits is {3, 2, 1, 0}. The corresponding integer is 3210. (1 integer)\r<BR>\n" );
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document.write( "Case 5: More than four digits<BR>\n" );
document.write( "If there are 5 distinct non-negative digits, the minimum sum would be $4 + 3 + 2 + 1 + 0 = 10$, which is greater than 6. So, we cannot have more than 4 digits.\r<BR>\n" );
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document.write( "The positive integers satisfying the conditions are 6, 51, 42, 321, and 3210.<BR>\n" );
document.write( "The number of such integers is $1 + 2 + 1 + 1 = 5$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{5}$ </SPAN>\n" );
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