1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Number of non negative integer solutions to this inequality

  1. Dec 1, 2015 #1
    1. The problem statement, all variables and given/known data
    How many non-negative integer solutions are there to the equation
    x1 + x2 + x3 + x4 + x5 < 11,
    (i)if there are no restrictions?
    (ii)How many solutions are there if x1 > 3?
    (iii)How many solutions are there if each xi < 3?

    2. Relevant equations
    N/A

    3. The attempt at a solution
    (i) inequality equivalent to equality x1 + x2 + x3 + x4 + x5 + x6 = 10
    (n+k-1)choose(k-1) = (10+6-1)choose(6-1) = 3003

    (ii) if x1 > 3 ------------> x2 + x3 + x4 + x5 < 8
    equality equivalent x2 + x3 + x4 + x5 +x6 = 7
    again (7+5-1)choose(4) = 330

    (iii)
    Its this part I'm not certain about

    like in (ii) let x1 instead be > 3 we find 7+5-1)choose(4) = 330

    now let x1 and x2 be > 3
    we find x3 + x4 + x5 < 5
    ----> x3 + x4 + x5 +x6 = 4
    (4+4-1)choose(4-1) =35

    finally let x1,x2 and x3 >3
    then x4 +x5 <2
    equality x4 + x5 + x6 =1
    (1+3-1)choose(2) = 3

    answer: 3003 - 35 - 330 - 3 = 2635

    any help would be appreciated.
     
  2. jcsd
  3. Dec 1, 2015 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You made a small error in (ii). If x1>3 then x1>=?
    (iii) is actually the easiest of the questions. If every term is under 3, and there are only 5 terms....
     
  4. Dec 1, 2015 #3
    is this correct?
    (ii)
    If x1>3 then x1 >= 4
    if x1 >= 4 ------------> x2 + x3 + x4 + x5 < 7
    equality equivalent x2 + x3 + x4 + x5 + x6 = 6
    (6+5-1)choose(5-1)
    (iii) since xi < 3
    each term can be 0,1 or 2
    so 3^5
     
  5. Dec 1, 2015 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Looks right to me.
     
  6. Dec 1, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You have part (i) wrong: the totality of non-negative integer solutions to ##x_1+x_2+x_3+x_4+x_5 < 11## is all the solutions to ##\sum x_i = 0## plus all the solutions to ##\sum x_i = 1##, plus ... plus all the solutions to ##\sum x_i = 9## plus all the solutions to ##\sum x_i = 10##. These all have sums < 11, as requested.

    The solutions to other parts will be similarly affected.
     
  7. Dec 1, 2015 #6

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Did the trick of introducing a sixth unknown to take up the slack not resolve that?
     
  8. Dec 1, 2015 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it does, but I missed that!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Number of non negative integer solutions to this inequality
  1. Negative integer trig (Replies: 11)

Loading...