The number of four digit number formed with the digits 1,1,2,2,3,4 is
Answers
Step-by-step explanation:
102 numbers of 4-digit numbers will be formed.
Case 1: Permutations of {1,2,3,4}
Total: 4! = 24 ways
Case 2: Permutations with 1 pair of identical digits:
Choose the 1 digit to repeat in 2 ways in C(2,1) = 2 ways
Choose the 2 digits that don't repeat in C(3,2) = 3 ways.
The number of permutation is 4!/2! = 24/2 = 12 ways
Total: 2×3×12 = 72 ways
Case 3: Permutations with 2 pairs of identical digit.
That's the permutations of 1,1,2,2
Total: 4!/(2!2!) = 24/(2·2) = 24/4 = 6
Total: 24+72+6 = 102 ways.
FYI, here are all 102 such 4-digit numbers:
1122 1123 1124 1132 1134 1142 1143 1212 1213 1214
1221 1223 1224 1231 1232 1234 1241 1242 1243 1312
1314 1321 1322 1324 1341 1342 1412 1413 1421 1422
1423 1431 1432 2112 2113 2114 2121 2123 2124 2131
2132 2134 2141 2142 2143 2211 2213 2214 2231 2234
2241 2243 2311 2312 2314 2321 2324 2341 2342 2411
2412 2413 2421 2423 2431 2432 3112 3114 3121 3122
3124 3141 3142 3211 3212 3214 3221 3224 3241 3242
3411 3412 3421 3422 4112 4113 4121 4122 4123 4131
4132 4211 4212 4213 4221 4223 4231 4232 4311 4312
4321 4322
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