document.write( "Question 866702: the no. of 4 digit numbers formed with the digits 1,1,2,2,3,4 \n" ); document.write( "
Algebra.Com's Answer #524925 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
the no. of 4 digit numbers formed with the digits 1,1,2,2,3,4
\n" ); document.write( "
\r\n" );
document.write( "Case 1: Permutations of {1,2,3,4}\r\n" );
document.write( "\r\n" );
document.write( "        Total:  4! = 24 ways\r\n" );
document.write( "\r\n" );
document.write( "Case 2: Permutations with 1 pair of identical digits:\r\n" );
document.write( "\r\n" );
document.write( "        Choose the 1 digit to repeat in 2 ways in C(2,1) = 2 ways          \r\n" );
document.write( "        Choose the 2 digits that don't repeat in C(3,2) = 3 ways.\r\n" );
document.write( "        The number of permutation is 4!/2! = 24/2 = 12 ways\r\n" );
document.write( "\r\n" );
document.write( "        Total: 2×3×12 = 72 ways\r\n" );
document.write( "\r\n" );
document.write( "Case 3: Permutations with 2 pairs of identical digit.\r\n" );
document.write( "\r\n" );
document.write( "That's the permutations of 1,1,2,2\r\n" );
document.write( "\r\n" );
document.write( "        Total: 4!/(2!2!) = 24/(2·2) = 24/4 = 6\r\n" );
document.write( "\r\n" );
document.write( "Total: 24+72+6 = 102 ways.\r\n" );
document.write( "\r\n" );
document.write( "FYI, here are all 102 such 4-digit numbers:\r\n" );
document.write( "\r\n" );
document.write( "1122 1123 1124 1132 1134 1142 1143 1212 1213 1214\r\n" );
document.write( "1221 1223 1224 1231 1232 1234 1241 1242 1243 1312\r\n" );
document.write( "1314 1321 1322 1324 1341 1342 1412 1413 1421 1422\r\n" );
document.write( "1423 1431 1432 2112 2113 2114 2121 2123 2124 2131\r\n" );
document.write( "2132 2134 2141 2142 2143 2211 2213 2214 2231 2234\r\n" );
document.write( "2241 2243 2311 2312 2314 2321 2324 2341 2342 2411\r\n" );
document.write( "2412 2413 2421 2423 2431 2432 3112 3114 3121 3122\r\n" );
document.write( "3124 3141 3142 3211 3212 3214 3221 3224 3241 3242\r\n" );
document.write( "3411 3412 3421 3422 4112 4113 4121 4122 4123 4131\r\n" );
document.write( "4132 4211 4212 4213 4221 4223 4231 4232 4311 4312\r\n" );
document.write( "4321 4322\r\n" );
document.write( "\r\n" );
document.write( "Edwin

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