document.write( "Question 605823: You are given the following seven-digit number: 4064465.\r
\n" ); document.write( "\n" ); document.write( "How many different seven-digit numbers can be made by rearranging the digits of 4064465? \n" ); document.write( "

Algebra.Com's Answer #381827 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
4064465
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "First we will do the problem as if we could tell the difference between\r\n" );
document.write( "the three 4's and the two 6's.\r\n" );
document.write( "\r\n" );
document.write( "Then, we could choose the 1st digit 6 ways.  (We cannot use 0)\r\n" );
document.write( "Then, we could choose the 2nd digit 6 ways.\r\n" );
document.write( "Then, we could choose the 3rd digit 5 ways.\r\n" );
document.write( "Then, we could choose the 4th digit 4 ways.\r\n" );
document.write( "Then, we could choose the 5th digit 3 ways.\r\n" );
document.write( "Then, we could choose the 6th digit 2 ways.\r\n" );
document.write( "Then, we could choose the 7th digit only 1 way.\r\n" );
document.write( "\r\n" );
document.write( "That would be 6·6·5·4·3·2·1 = 4320 ways.\r\n" );
document.write( "\r\n" );
document.write( "If the three 4's and the two 6's were distinguishable, for instance, if\r\n" );
document.write( "the like digits were different colors, then 4320 would be the final \r\n" );
document.write( "answer.  However, they're not distinguishable, so let's look at a random \r\n" );
document.write( "sample permutation, say, this one:\r\n" );
document.write( "\r\n" );
document.write( "6045464.\r\n" );
document.write( "\r\n" );
document.write( "The answer 6·6·5·4·3·2·1 = 4320 counts all 12 of the following\r\n" );
document.write( "arrangements separately:\r\n" );
document.write( "\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "6045464\r\n" );
document.write( "\r\n" );
document.write( "But we cannot tell them apart.  So 6·6·5·4·3·2·1 = 4320 counte EVERY\r\n" );
document.write( "arrangement 12 times too many, since we cannot tell any of those 12 from\r\n" );
document.write( "6045464.  So we must divide 4320 by 12, and get the answer 360.  \r\n" );
document.write( "\r\n" );
document.write( "But how did we know to divide by 12?  That's because there are 3! ways\r\n" );
document.write( "to place the 3 different colored 3's in the above colored list and for \r\n" );
document.write( "each of those 3 different colored 3's, there are 2! ways to place the\r\n" );
document.write( "colored 2's.  So we divide by 3!2! = 6×2 = 12 \r\n" );
document.write( "\r\n" );
document.write( "So the answer is\r\n" );
document.write( "\r\n" );
document.write( " =  = 360.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );