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Mathematics Dictionary
Dr. K. G. Shih
Arrangement of 4 digits


  • Q01 | - Arrangements of two digit number from 1, 2, 3, 4
  • Q02 | - Arrangements of three digit number from 1, 2, 3, 4
  • Q03 | - Arrangements of four digit number from 1, 2, 3, 4
  • Q04 | - Formula

    • Q01. Arrangements of two digits number from 1, 2, 3, 4

    Case 1 : Same digit is not included

    • The answer is P(4, 2) = 4*3 = 12
    • Iluustration
      • 43, 42, 41
      • 34, 31, 32
      • 21, 23, 24
      • 12, 13, 14
    • Hence the arrangement is 12
    Case 2 : Same digit can be used

    • The answer is 4^2 = 16
    • Different digits : 12
    • Same digits used
      • 44
      • 33
      • 22
      • 11
      • It has 4 arrangement
    • Hence the arragement is 4 + 12 = 16 = 4^2

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    Q02. Arrangements of three digits number from 1, 2, 3, 4

    Case 1 : All three digits are differenct
    • Arrangement is P(4, 3) = 4*3*2 = 24
    • Illustration
      • 431, 432, 423, 421, 412, 413
      • 341, 342, 324, 342, 314, 341
      • 234, ...
      • 123, ...
    • It has 24 arrangement
    Case 2 : Two same digits are included
    • Three digits are different : Arrangement is P(4, 3) = 4*3*2*1 = 24
    • Two same digits
      • 433, 343, 334, 422, 242, 224, 411, 141, 114
      • 344, 434, 333, 322, 232, 223, 311, 131, 113
      • 244, 424, 442, 233, 323, 332, 211, 121, 112
      • 144, 141, 441, 133, 313, 331, 122, 212, 221
      • It has 36 arrangement for two same digts
    • Three same digits are included
      • 444
      • 333
      • 222
      • 111
      • It has 4 arrangnement for three digits same
    • Total arrangement = 24 + 36 + 4 = 64 = 4^3
    Conclusion
    • Four digits all different ....... Arrangement = 4! = 24
    • Same digits can be used .......... Arrangement = 4^3 = 64
    • Same two digits can be used ...... Arrangement = 4^3 - 4 = 60

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    Q03. Arrangements of four digit number from 1, 2, 3, 4

    All four digits are different
    • The arragnement P(4,4) = 4*3*2*1 = 24
    • Illustration
      • 1234, 1243, 1342, 1324, 1423, 1432
      • 2341, 2314, 2413, 2431, 2134, 2143
      • 3412, ....
      • 4123, ....
      • Total arragment is 24
    All four digits can be reapted
    • The arrangement is 4^4 = 256

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    Q04. Formula

    Number with m digits from n digits
    • If all m digits are different, the arrangement is P(n, m)
    • If m digits can be repeated, the arrangment is n^m

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