-1

how many solutions $(x_1,x_2,x_3,x_4)$ are there to the equation $x_1+x_2+x_3+x_4=18$, $1\leq x_1<x_2<x_3<x_4\leq 9$, $x_1,x_2,x_3,x_3\in\mathbb{N}$. Thanks so much!

dnmath
  • 7
  • Try using generating functions, for example. There are plenty of examples on MSE, like this or this. – rtybase Nov 10 '19 at 09:01
  • @rty the first of those two questions doesn't have the condition that the $x_i$ are increasing. – Gerry Myerson Nov 10 '19 at 09:40
  • @GerryMyerson my argument is to use generating functions and I provided a few examples. What is not related to the question in your opinion? – rtybase Nov 10 '19 at 09:49
  • @rty how do you incorporate the condition $x_1<x_2<x_3<x_4$ into the generating function? Well, maybe you can, but there's nothing like that in the first example you link to, is there? – Gerry Myerson Nov 10 '19 at 10:37
  • 1
    @GerryMyerson my intent was not to solve OP's problem, but provide him with sufficient material to learn (possibly) a new technique. The 1st example is a very simple one (the only relation to the question is $=18$), the 2nd example contains a technique for $1\leq x \leq y \leq z$ and a link to this. – rtybase Nov 10 '19 at 10:42
  • Welcome to MathSE. Please edit your question to show what you have attempted and explain where you are stuck. – N. F. Taussig Nov 10 '19 at 14:59

1 Answers1

0

Here are all solutions: $$(\text{x1}=1\land \text{x2}=2\land \text{x3}=6\land \text{x4}=9)\lor (\text{x1}=1\land \text{x2}=2\land \text{x3}=7\land \text{x4}=8)\lor (\text{x1}=1\land \text{x2}=3\land \text{x3}=5\land \text{x4}=9)\lor (\text{x1}=1\land \text{x2}=3\land \text{x3}=6\land \text{x4}=8)\lor (\text{x1}=1\land \text{x2}=4\land \text{x3}=5\land \text{x4}=8)\lor (\text{x1}=1\land \text{x2}=4\land \text{x3}=6\land \text{x4}=7)\lor (\text{x1}=2\land \text{x2}=3\land \text{x3}=4\land \text{x4}=9)\lor (\text{x1}=2\land \text{x2}=3\land \text{x3}=5\land \text{x4}=8)\lor (\text{x1}=2\land \text{x2}=3\land \text{x3}=6\land \text{x4}=7)\lor (\text{x1}=2\land \text{x2}=4\land \text{x3}=5\land \text{x4}=7)\lor (\text{x1}=3\land \text{x2}=4\land \text{x3}=5\land \text{x4}=6)$$

Dr. Sonnhard Graubner
  • 95,283