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Mathematics Dictionary
Dr. K. G. Shih
Arrange number n as 4 integers


  • Q01 | - Arrange 4 in terms of 4 integers
  • Q02 | - Arrange 5 in terms of 4 integers
  • Q03 | - Arrange 6 in terms of 4 integers
  • Q04 | - Arrange 7 in terms of 4 integers
  • Q05 | - Conclusion

    • Q01. Find arrangement if 4 is expressed in terms of 4 integers

    An integer n and n = n1 + n2 + n2 + n4, How many arrangement

    Note
    • Four integers n1, n2, n3, n4 are positive (no zero)
    Projecture a formula for n = 4
    • n1 = 1, n2 = 1, n3 = 1 and n4 = 1
    • We have n = 1 + 1 + 1 + 1 = 4
    • It has only one arragement.
    • It is the case (n - 3) and has 1 arrangement

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    Q02. Find arrangement if 5 is expressed in terms of 4 integers

    Note
    • Four integers n1, n2, n3, n4 are positive (no zero)
    Projecture a formula for n = 5
    • Number (n-3) stands at 4th place
      • Number 2 at 4th place
      • n = 1 + 1 + 1 + 2 = 4
      • It has only one arragement if 2 stands at 3rd place
      • It is the case (n - 3) at 4th place has 1 arrangement
    • Number (n-4) stands at 4th place
      • Number 1 at 4th place
      • 1 + 1 + 2 + 1 = 5
      • 1 + 2 + 1 + 1 = 5
      • 2 + 1 + 1 + 1 = 5
      • It is the case for (n-4) at 4th place
      • Hence it has 3 arragements
    • Hence there are 1 + 3 = 4 arrangement
    • It has m terms : m = (n - 3) = 2
    • Sum = m*(m + 1)*(m + 2)/6
    • Hence arrangement = (n - 1)*(n - 2)*(n - 3)/6 = 4

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    Q03. Find arrangement if 6 is expressed in terms of 4 integers

    Note
    • Three integers n1, n2, n3, n4 are positive (no zero)
    Projecture a formula : n = 6
    • One number is (n - 3) = 3 : it is (n - 3) = 3 at 4th place
      • Number 3 at 4th place
      • 1 + 1 + 1 + 3 = 6
      • It hsa 1 arrangement
    • One number is (n - 4) = 2 : it is (n - 4) = 2 at 4th place
      • Number 2 at 4th place
      • 1 + 1 + 2 + 2 = 6
      • 1 + 2 + 1 + 2 = 6
      • 2 + 1 + 1 + 2 = 6
      • It has 3 arrangement
    • One number is (n - 5) = 1 : it is (n - 5) = 1 at 4th place
        >li> Number 1 at 4th place
      • 1 + 1 + 3 + 1 = 6
      • 1 + 3 + 1 + 1 = 6
      • 3 + 1 + 1 + 1 = 6
      • 1 + 2 + 2 + 1 = 6
      • 2 + 1 + 2 + 1 = 6
      • 2 + 2 + 1 + 1 = 6
      • It has 6 arrangement
    • Hence arrangement is 1 + 3 + 6 = 10
    Conclusion
    • Number of terms : m = (n - 3) = 3 if n = 6
    • Sum = m*(m + 1)*(m + 2)/6 = (n - 3)*(n - 2)*(n - 1)/2
    • Hence arranement = (6 - 3)*(6 - 2)*(6 - 1)/6 = (3*4*5)/6 = 10
    • It is C(n-1, 3) = (n - 1)*(n - 2)*(n - 3)/(3!)

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    Q04. Find arrangement if 7 is expressed in terms of 4 integers

    Note
    • Three integers n1, n2, n3, n4 are positive (no zero)
    Projecture a formula for n = 7
    • One number is (n - 3) = 4 : it is (n - 3) at 4th place
      • Number 4 stands at 4th place
      • n = 1 + 1 + 1 + 4 = 7. Only one arrangement
    • One number is (n - 4) = 3 : it is (n - 4) at 4th place
      • Number 3 stands at 4th place
      • 1 + 1 + 2 + 3 = 7
      • 1 + 2 + 1 + 3 = 7
      • 2 + 1 + 1 + 3 = 7
      • It has 3 arrangements
    • One number is (n - 5) = 2 : it is (n - 5) at 4th palce
      • Number 2 stands at 4th place
      • 1 + 1 + 3 + 2 = 7
      • 1 + 3 + 1 + 2 = 7
      • 3 + 1 + 1 + 2 = 7
      • 1 + 1 + 2 + 2 = 7
      • 1 + 2 + 1 + 2 = 7
      • 2 + 1 + 1 + 2 = 7
      • It has 6 arrangements
    • One number is (n - 6) = 1 : it is (n - 6) at 4th place
      • Number 1 stands at 4th place
      • 1 + 1 + 4 + 1 = 7
      • 1 + 2 + 3 + 1 = 7
      • 2 + 1 + 3 + 1 = 7
      • 1 + 3 + 2 + 1 = 7
      • 3 + 1 + 2 + 1 = 7
      • 1 + 4 + 1 + 1 = 7
      • 4 + 1 + 1 + 1 = 7
      • 2 + 3 + 1 + 1 = 7
      • 3 + 2 + 1 + 1 = 7
      • 2 + 2 + 2 + 1 = 7
      • It has 10 arrangenments
    • Total = 1 + 3 + 5 + 10 = 20
    Conclusion
    • Hence arranements = 1 + 3 + 6 + 10 = 20
    • Number of terms m = (n - 3) = 4
    • Sum = m*(m + 1)/2 = (n - 1)*(n - 2)*(n - 3)/6
    • It is C(n-1, 3) = (n - 1)*(n - 2)*(n - 3)/(3!)

    Go to Begin

    Q05. Conclusion

    Conclusion
    • For (n - 3), it has 1 arrangement
    • For (n - 4), it has 3 arrangements
    • For (n - 5), it has 6 arrangements
    • For (n - 6), it has 10 arrangements
    • Etc.
    • Number of terms : m = (n - 3)
    • Sum = m*(m + 1)*(m + 2)/6
    • Hence total arrangements = 1 + 3 + 6 + ... = (n - 3)*(n - 2)*(n - 1)/6
    • It is C(n-1, 3) = (n - 1)*(n - 2)*(n - 3)/(3!)

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