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Mathematics Dictionary
Dr. K. G. Shih
Arrangement of 3 digits


  • Q01 | - Arrangements of two digit number from 1, 2, 3
  • Q02 | - Arrangements of three digit number from 1, 2, 3
  • Q03 | - Probability : Number less than 200 of numbers fromed by 1, 2, 3
  • Q04 | - Formula

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

    Case 1 : Same digit is not included

    • The answer is P(3, 2) = 3*2 = 6
    • Iluustration
      • 32, 31
      • 23, 21
      • 12, 13
    • Hence the arrangement is 6
    Case 2 : Same digit can be used

    • The answer is 3^2 = 9
    • Iluustration of different digits
      • 32, 31
      • 23, 21
      • 12, 13
      • It ha 6 arrangement
    • Illustration of two same digits
      • 33
      • 22
      • 11
      • It has 3 arrangement
    • Hence the arrangement is 3 + 6 = 9 = 3^2

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

    Case 1 : All digits are differenct
    • Arrangement is P(3, 3) = 3! = 3*2*1 = 6
    • Illustration
      • 321 312
      • 231 213
      • 123 123
    • It has 6 arrangement
    Case 2 : Two same digits are included
    • Three digits are different : Arrangement is P(3, 3) = 3! = 3*2*1 = 6
    • Illustration
      • 321 312
      • 231 213
      • 123 123
      • It has 6 arrangement for all digits are different
    • Two same digits are included
      • 322, 232, 223
      • 311, 131, 113
      • 233, 323, 332
      • 211, 121, 112
      • 133, 131, 113
      • 122, 212, 221
      • It has 18 arrangement for two digits same
    • Three same digits are included
      • 333
      • 222
      • 111
      • It has 3 arrangement for three digits same
    • Total arrangement = 3 + 6 + 18 = 27
    Conclusion
    • Three digits all different ....... Arrangement = 3! = 6
    • Same digits can be used .......... Arrangement = 3^3 = 27
    • Same two digits can be used ...... Arrangement = 3^3 - 3 = 24
    • Formula : Arrangement = 3^3 - 3

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    Q03. Three digit numbers from digits 1, 2, 3

    Find probability to get number less than 200
    • Total sample sapce is S = 27
    • Since number is less than 200, hence 2 and 3 will not in hundred place
    • The arragnement
      • 123, 132
      • 122, 133
      • 113, 131
      • 112, 121
      • 111
      • Total arragment is 9
    • Hence probability is 9/27 = 1/3

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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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