document.write( "Question 1146125: If all digits 0,1,2,3,4,5,6,7,8,9 are rotated through a 180 degrees ,the
\n" ); document.write( "digits 0,1, and 8 do not change meaning, the digit 6 becomes 9, and 9
\n" ); document.write( "becomes 6. The remaining digits lose their meaning. a palidium is a positive
\n" ); document.write( "whole number that remains the same after a 180 degree rotation(for example, 1691 or 8008). How many numbers between 1000 and 9999(inclusive) are
\n" ); document.write( "palidiums? \n" ); document.write( "

Algebra.Com's Answer #767412 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "

First we see how many ways we can have the outer two digits. \r\n" );
document.write( "They can be\r\n" );
document.write( "\r\n" );
document.write( "1 _ _ 1\r\n" );
document.write( "\r\n" );
document.write( "8 _ _ 8\r\n" );
document.write( "\r\n" );
document.write( "6 _ _ 9\r\n" );
document.write( "\r\n" );
document.write( "9 _ _ 6\r\n" );
document.write( "\r\n" );
document.write( "So we can pick the outer digits 4 different ways.\r\n" );
document.write( "\r\n" );
document.write( "Next we see how many ways we can have the middle two digits. \r\n" );
document.write( "They can be\r\n" );
document.write( "\r\n" );
document.write( "_ 0 0 _\r\n" );
document.write( "\r\n" );
document.write( "_ 1 1 _\r\n" );
document.write( "\r\n" );
document.write( "_ 8 8 _\r\n" );
document.write( "\r\n" );
document.write( "_ 6 9 _\r\n" );
document.write( "\r\n" );
document.write( "_ 9 6 _\r\n" );
document.write( "\r\n" );
document.write( "So we can pick middle two digits 5 different ways. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So for each of the 4 ways we can pick the outer two digit,\r\n" );
document.write( "we can pick the inner two digits 5 ways.\r\n" );
document.write( "So there are 4×5 or 20 ways to have a palidum.\r\n" );
document.write( "\r\n" );
document.write( "Here are all 20 palidia:\r\n" );
document.write( "\r\n" );
document.write( " 1. 1 0 0 1\r\n" );
document.write( " 2. 8 0 0 8\r\n" );
document.write( " 3. 6 0 0 9\r\n" );
document.write( " 4. 9 0 0 6\r\n" );
document.write( " 5. 1 1 1 1\r\n" );
document.write( " 6. 8 1 1 8\r\n" );
document.write( " 7. 6 1 1 9\r\n" );
document.write( " 8. 9 1 1 6\r\n" );
document.write( " 9. 1 8 8 1\r\n" );
document.write( "10. 8 8 8 8\r\n" );
document.write( "11. 6 8 8 9\r\n" );
document.write( "12. 9 8 8 6\r\n" );
document.write( "13. 1 6 9 1\r\n" );
document.write( "14. 8 6 9 8\r\n" );
document.write( "15. 6 6 9 9\r\n" );
document.write( "16. 9 6 9 6\r\n" );
document.write( "17. 1 9 6 1\r\n" );
document.write( "18. 8 9 6 8\r\n" );
document.write( "19. 6 9 6 9\r\n" );
document.write( "20. 9 9 6 6 \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );