document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208643>Question 1208643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We call a number cozy if every digit in the number is either a $3$ or next to a $3.$  For example, the numbers $333,$ $83,$ $303,$ and $3773$ are all cozy, but the numbers $32423,$ $786,$ $340,$ and $3999$ are not cozy.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "How many positive $10$-digit numbers are cozy? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847652 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=847652\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down the problem.\r<BR>\n" );
document.write( "\n" );
document.write( "**Observation 1:** A cozy number must contain at least one $3$. \r<BR>\n" );
document.write( "\n" );
document.write( "**Observation 2:** If a $3$ is in a position, the adjacent positions must also contain a $3$ to make the number cozy. \r<BR>\n" );
document.write( "\n" );
document.write( "**Observation 3:** A cozy number of length 10 can have a maximum of 5 $3$'s. \r<BR>\n" );
document.write( "\n" );
document.write( "**Case 1: One 3**<BR>\n" );
document.write( "* The $3$ can be in any of the 10 positions.<BR>\n" );
document.write( "* For each position, there are 9 choices for the other digits (any digit except $3$).<BR>\n" );
document.write( "* So, there are $10 \times 9^9$ possibilities.\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 2: Two 3's**<BR>\n" );
document.write( "* The $3$'s must be adjacent.<BR>\n" );
document.write( "* There are 9 such pairs of positions (1-2, 2-3, ..., 9-10).<BR>\n" );
document.write( "* For the other 8 positions, there are 9 choices for each.<BR>\n" );
document.write( "* So, there are $9 \times 9^8$ possibilities.\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 3: Three 3's**<BR>\n" );
document.write( "* The $3$'s must be in three consecutive positions.<BR>\n" );
document.write( "* There are 8 such triplets of positions.<BR>\n" );
document.write( "* For the other 7 positions, there are 9 choices for each.<BR>\n" );
document.write( "* So, there are $8 \times 9^7$ possibilities.\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 4: Four 3's**<BR>\n" );
document.write( "* The $3$'s must be in two pairs of consecutive positions.<BR>\n" );
document.write( "* There are ${8 \choose 2}$ ways to choose the positions for the pairs.<BR>\n" );
document.write( "* For the other 6 positions, there are 9 choices for each.<BR>\n" );
document.write( "* So, there are ${8 \choose 2} \times 9^6$ possibilities.\r<BR>\n" );
document.write( "\n" );
document.write( "**Case 5: Five 3's**<BR>\n" );
document.write( "* The $3$'s must be in five consecutive positions.<BR>\n" );
document.write( "* There are 6 such quintuplets of positions.<BR>\n" );
document.write( "* For the other 5 positions, there are 9 choices for each.<BR>\n" );
document.write( "* So, there are $6 \times 9^5$ possibilities.\r<BR>\n" );
document.write( "\n" );
document.write( "Adding up all the cases, we get the total number of cozy 10-digit numbers:\r<BR>\n" );
document.write( "\n" );
document.write( "$10 \times 9^9 + 9 \times 9^8 + 8 \times 9^7 + {8 \choose 2} \times 9^6 + 6 \times 9^5$ <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );