Results 1 to 4 of 4

Math Help - # of integer solutions

  1. #1
    Super Member
    Joined
    Feb 2008
    Posts
    535

    # of integer solutions

    Here's an example problem:
    Find the number of integer solutions to the equation:
    x1 + x2 + x3 + x4 = 20
    with the restrictions:
    x1 >=1; x2 >=3; x3 >= -3; x4 >= 0

    To do this we create new variables as follows:

    y1 = x1 - 1 >=0; y2 = x2 - 3 >=0; y3 = x3 + 3 >=0; y4 = x4 - 0 >=0

    y1 + y2 + y3 + y4 = (x1-1) + (x2-3) + (x3+3) + (x4-0) = x1 + x2 + x3 + x4 -1 = 20 -1 = 19.

    Thus, 19+4-1 C 4-1 = 22C3 integer solutions.

    How do I go about it if my restrictions are as follows: (less than instead of greater)
    x1 <= 8; x2 <=10; x3 <= 12; x4 <=5

    Thansk
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie kira's Avatar
    Joined
    Nov 2010
    Posts
    17
    use the same method as u have done as above choosing y1= 8-x1 ad similarly y2,y3,y4

    at the end then u will have y1+y2+y3+y4=35-20=15

    it has integer solutions 15+4-1C4-1 = 15C3
    Last edited by kira; March 2nd 2011 at 07:26 PM. Reason: integer solutions
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Feb 2008
    Posts
    535
    Thanks!! Now, what would I do differently if there were NO restrictions on x1,...,x4?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie kira's Avatar
    Joined
    Nov 2010
    Posts
    17
    Quote Originally Posted by jzellt View Post
    Thanks!! Now, what would I do differently if there were NO restrictions on x1,...,x4?
    well it would be obvious that there can be infinitely many such solutions in such a case

    infact the only time when u can give the number of silutions is when u have restraints on all the 'n' (out of the 'n' numbers being added) being restricted or at the most '(n-1)' numbers (out of the n numbers being added) being restricted

    even if they restrict '(n-2)' numbers out of the n numbers, then u can always choose the remaining 2 numbers in infinitely many ways to get the desired result
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. No integer Solutions
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: July 16th 2011, 03:01 AM
  2. Replies: 2
    Last Post: May 8th 2010, 10:59 PM
  3. Integer Solutions?
    Posted in the Number Theory Forum
    Replies: 44
    Last Post: September 14th 2009, 09:11 PM
  4. Integer Solutions
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: July 31st 2009, 09:18 AM
  5. Integer solutions to a^2+b^2=c^3
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 11th 2009, 07:48 AM

Search Tags


/mathhelpforum @mathhelpforum