Counter
Mathematics Dictionary
Dr. K. G. Shih
Arrange number n as 5 integers


  • Q01 | - Arrange 5 in terms of 5 integers
  • Q02 | - Arrange 6 in terms of 5 integers
  • Q03 | - Arrange 7 in terms of 5 integers
  • Q04 | - Arrange 8 in terms of 5 integers
  • Q05 | - Conclusion

    • Q01. Find arrangement if 5 is expressed in terms of 5 integers

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

    Note
    • Four integers n1, n2, n3, n4, n5 are positive (no zero)
    Projecture a formula for n = 5
    • n1 = 1, n2 = 1, n3 = 1, n4 = 1 n5 = 1
    • We have n = 1 + 1 + 1 + 1 + 1 = 5
    • It has only one arragement.
    • It is the case (n - 4) and has 1 arrangement

    Go to Begin

    Q02. Find arrangement if 6 is expressed in terms of 5 integers

    Note
    • Four integers n1, n2, n3, n4, n5 are positive (no zero)
    Projecture a formula for n = 6
    • Number (n-4) stands at 5th place
      • Number 2 at 5th place
      • n = 1 + 1 + 1 + 1 + 2 = 6
      • It has only one arragement if 2 stands at 5th place
      • It is the case (n - 4) at 4th place has 1 arrangement
    • Number (n-5) stands at 4th place
      • Number 1 at 5th place
      • 1 + 1 + 1 + 2 + 1 = 6
      • 1 + 1 + 2 + 1 + 1 = 6
      • 1 + 2 + 1 + 1 + 1 = 6
      • 2 + 1 + 1 + 1 + 1 = 6
      • It is the case for (n-5) at 4th place
      • Hence it has 4 arragements
    • Hence there are 1 + 4 = 5 arrangement
    • It has m terms : m = (n - 3) = 2
    • Sum = m*(m + 1)*(m + 2)/6
    • Hence arrangement = (n - 1)*(n - 2)*(n - 3)*(n - 4)/24 = 5

    Go to Begin

    Q03. Find arrangement if 7 is expressed in terms of 5 integers

    Note
    • Three integers n1, n2, n3, n4, n5 are positive (no zero)
    Projecture a formula : n = 7
    • One number is (n - 4) = 3 : it is (n - 4) = 3 at 5th place
      • Number 3 at 5th place
      • 1 + 1 + 1 + 1 + 3 = 7
      • It has 1 arrangement
    • One number is (n - 5) = 2 : it is (n - 5) = 2 at 5th place
      • Number 2 at 5th place
      • 1 + 1 + 1 + 2 + 2 = 7
      • 1 + 1 + 2 + 1 + 2 = 7
      • 1 + 2 + 1 + 1 + 2 = 7
      • 2 + 1 + 1 + 1 + 2 = 7
      • It has 4 arrangement
    • One number is (n - 6) = 1 : it is (n - 5) = 1 at 5th place
      • Number 1 at 5th place
      • 1 + 1 + 1 + 3 + 1 = 7
      • 1 + 1 + 3 + 1 + 1 = 7
      • 1 + 3 + 1 + 1 + 1 = 7
      • 3 + 1 + 1 + 1 + 1 = 7
      • 1 + 1 + 2 + 2 + 1 = 7
      • 1 + 2 + 2 + 1 + 1 = 7
      • 2 + 1 + 2 + 1 + 1 = 7
      • 2 + 2 + 1 + 1 + 1 = 7
      • 1 + 2 + 1 + 2 + 1 = 7
      • 2 + 1 + 1 + 2 + 1 = 7
      • It has 10 arrangement
    • Hence arrangement is 1 + 4 + 10 = 15
    Conclusion
    • Number of terms : m = (n - 4) = 3 if n = 7
    • It has m = (n - 4) terms
    • Sum = m*(m + 1)*(m + 2)*(m + 3)/(4!) = (n - 4)*(n - 3)*(n - 2)*(n - 1)/24
    • Hence arranement =(7-4)*(7-3)*(7-2)*(7-1)/24 = (3*4*5*6)/24 = 15
    • It is C(n-1, 4) = (n - 1)*(n - 2)*(n - 3)*(n - 4)/(4!)

    Go to Begin

    Q04. Find arrangement if 8 is expressed in terms of 5 integers

    Note
    • Three integers n1, n2, n3, n4, n5 are positive (no zero)
    Projecture a formula for n = 8
    • One number is (n - 4) = 4 : it is (n - 4) at 5th place
      • Number 4 stands at 5th place
      • n = 1 + 1 + 1 + 1 + 4 = 8. Only one arrangement
    • One number is (n - 5) = 3 : it is (n - 5) at 5th place
      • Number 3 stands at 5th place
      • 1 + 1 + 1 + 2 + 3 = 8
      • 1 + 1 + 2 + 1 + 3 = 8
      • 1 + 2 + 1 + 1 + 3 = 8
      • 2 + 1 + 1 + 1 + 3 = 8
      • It has 4 arrangements
    • One number is (n - 6) = 2 : it is (n - 6) at 5th palce
      • Number 2 stands at 5th place
      • 1 + 1 + 1 + 3 + 2 = 8
      • 1 + 1 + 3 + 1 + 2 = 8
      • 1 + 3 + 1 + 1 + 2 = 8
      • 3 + 1 + 1 + 1 + 2 = 8
      • 1 + 1 + 2 + 2 + 2 = 8
      • 1 + 2 + 1 + 2 + 2 = 8
      • 2 + 1 + 1 + 2 + 2 = 8
      • 1 + 2 + 2 + 1 + 2 = 8
      • 2 + 2 + 1 + 1 + 2 = 8
      • 2 + 1 + 2 + 1 + 2 = 8
      • It has 10 arrangements
    • One number is (n - 6) = 1 : it is (n - 6) at 4th place
      • Number 1 stands at 4th place
      • 1 + 1 + 1 + 4 + 1 = 8
      • 1 + 1 + 4 + 1 + 1 = 8
      • 1 + 4 + 1 + 1 + 1 = 8
      • 4 + 1 + 1 + 1 + 1 = 8
      • Etc.
      • It has 20 arrangenments
    • Total = 1 + 3 + 5 + 10 = 20
    Conclusion
    • Hence arranements = 1 + 4 + 10 + 20 = 35
    • Number of terms m = (n - 4) = 4
    • Sum = m*(m + 1)/2 = (n - 1)*(n - 2)*(n - 3)(n - 4)/24
    • It is C(n-1, 4) = (n - 1)*(n - 2)*(n - 3)*(n - 4)/(4!) = (7*6*5*4)/24 = 35

    Go to Begin

    Q05. Conclusion

    Conclusion
    • For (n - 4), it has 1 arrangement
    • For (n - 5), it has 4 arrangements
    • For (n - 6), it has 10 arrangements
    • For (n - 7), it has 25 arrangements
    • Etc.
    • Number of terms : m = (n - 4)
    • Sum = m*(m + 1)*(m + 2)*(m + 3)/(4!)
    • Hence arrangements = 1 + 4 + 10 + ... = (n - 4)*(n - 3)*(n - 2)*(n - 1)/(4!)
    • It is C(n-1, 4) = (n - 1)*(n - 2)*(n - 3)*(n - 4)/(4!)

    Go to Begin
    Show Room of MD2002 Contact Dr. Shih Math Examples Room
    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
    1