document.write( "Question 706693: How many seven-digit telephone numbers are there that begin with 347 and contain at least one 0 and at least one 1 ? \n" ); document.write( "
Algebra.Com's Answer #435351 by Edwin Parker(36)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Since they all start with 347, we only need to consider\r\n" );
document.write( "the last 4 digits.\r\n" );
document.write( "\r\n" );
document.write( "There are 6 cases to consider.\r\n" );
document.write( "\r\n" );
document.write( "Let x = the exact number of zeros, \r\n" );
document.write( "Let y = the exact number of ones\r\n" );
document.write( "Then the exact number of other digits (2 through 9) is 4-x-y\r\n" );
document.write( "\r\n" );
document.write( "     Number    Number    Number  \r\n" );
document.write( "       of        of        of\r\n" );
document.write( "      zeros     ones     others\r\n" );
document.write( "Case    x        y        4-x-y\r\n" );
document.write( "--------------------------------\r\n" );
document.write( "A       1        1         2\r\n" );
document.write( "B       1        2         1\r\n" );
document.write( "C       1        3         0\r\n" );
document.write( "D       2        1         2\r\n" );
document.write( "E       2        2         0\r\n" );
document.write( "F       3        1         0\r\n" );
document.write( "\r\n" );
document.write( "Cases B and D are the same.\r\n" );
document.write( "Cases C and F are the same.\r\n" );
document.write( "\r\n" );
document.write( "In each of those 6 cases,\r\n" );
document.write( "\r\n" );
document.write( "There are \"4 choose x\" positions to place the 0's\r\n" );
document.write( "\r\n" );
document.write( "There are then \"4-x choose y\" positions to place the 1's\r\n" );
document.write( "\r\n" );
document.write( "There are then 84-x-y ways to choose the others.\r\n" );
document.write( "\r\n" );
document.write( "That's C(4,x)·C(4-x,y)·84-x-y for each case.\r\n" );
document.write( "\r\n" );
document.write( "Number of cases of A = C(4,1)C(3,1)82 = 4·3·8·64 = 768\r\n" );
document.write( "Number of cases of B = C(4,1)C(3,2)81 = 4·3·8    =  96\r\n" );
document.write( "Number of cases of C = C(4,1)C(3,3)80 = 4·1·1    =   4\r\n" );
document.write( "Number of cases of D = same as for case B         = 96\r\n" );
document.write( "Number of cases of E = C(4,2)C(2,2)80 = 6·1·1    =   6\r\n" );
document.write( "Number of cases of F = same as for case C         =  4\r\n" );
document.write( "------------------------------------------------------\r\n" );
document.write( "Total                                              974\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );