# Find the number of integer solutions

• November 14th 2011, 02:34 PM
tinabaker
Find the number of integer solutions
Find the number of integer solutions x1+x2+x3=20 with x1≥0 x2≥1 x3≥1
is this C(20,2)?
Thank you.
• November 14th 2011, 02:54 PM
TKHunny
Re: Find the number of integer solutions
How did you get that result?

Take y2 = x2 - 1 and y3 = x3 - 1

Rephrase to x1 + (y2 + 1) + (y3 + 1) = 20 or x1 + y2 + y3 = 18

And out pops C(18+2,2). I think you have it!
• November 14th 2011, 03:03 PM
Plato
Re: Find the number of integer solutions
Quote:

Originally Posted by tinabaker
Find the number of integer solutions x1+x2+x3=20 with x1≥0 x2≥1 x3≥1
is this C(20,2)?

When posting please do not use special fonts.
The number of way to put N 1's into K variables is $\tbinom{N+K-1}{N}$.
If we want $x_2\ge 1~\&~x_3\ge 1$ we can think of putting a 1 into each of those two variables.
So is the answer $\tbinom{18+3-1}{18}~?$
• November 14th 2011, 03:06 PM
tinabaker
Re: Find the number of integer solutions
thank you for your help :)
• November 24th 2011, 05:52 PM
CaramelCardinal
Re: Find the number of integer solutions
If you require all x's to be 0 or greater, the equation is ((N-1)C(K-1)).