document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000962>Question 1000962</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a pair of dice is rolled, in how many ways are the two numbers different? If three dice are rolled, in how many ways are at least two of the dice the same? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618285 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618285\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The dice have 36 outcomes.<BR>\n" );
document.write( "6 of them are the same: 1,1;2,2;3,3;4,4;5,5;6,6<BR>\n" );
document.write( "the other 30 are not.<BR>\n" );
document.write( "For 3 dice, there are 216 outcomes.<BR>\n" );
document.write( "at least two means two or three are the same.<BR>\n" );
document.write( "three occurs 36 in 216 times.  That is (1/6)(1/6)(1/6)*6 for each number (1/6)-111,222,333,444,555,666<BR>\n" );
document.write( "Suppose the first die is a 1: two occur (1/6)(1/6)(5/6)*3, because there are 3 ways the two can occur.<BR>\n" );
document.write( "That is 15/216.  But that can occur for each of the six numbers.  The probability is 90/216 that exactly two will occur, and the probability 96/216, or 6/13.<BR>\n" );
document.write( "111 <BR>\n" );
document.write( "112<BR>\n" );
document.write( "113<BR>\n" );
document.write( "114<BR>\n" );
document.write( "115<BR>\n" );
document.write( "116<BR>\n" );
document.write( "121<BR>\n" );
document.write( "122<BR>\n" );
document.write( "123<BR>\n" );
document.write( "124<BR>\n" );
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document.write( "126<BR>\n" );
document.write( "131<BR>\n" );
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document.write( "133<BR>\n" );
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document.write( "135<BR>\n" );
document.write( "136<BR>\n" );
document.write( "141<BR>\n" );
document.write( "142<BR>\n" );
document.write( "143<BR>\n" );
document.write( "144<BR>\n" );
document.write( "145<BR>\n" );
document.write( "146<BR>\n" );
document.write( "151<BR>\n" );
document.write( "152<BR>\n" );
document.write( "153<BR>\n" );
document.write( "154<BR>\n" );
document.write( "155<BR>\n" );
document.write( "156<BR>\n" );
document.write( "161<BR>\n" );
document.write( "162<BR>\n" );
document.write( "163<BR>\n" );
document.write( "164<BR>\n" );
document.write( "165<BR>\n" );
document.write( "166 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );