# Homework help

• Oct 13th 2009, 09:59 AM
LAURA.GOODMAN
Homework help
Find the number of integer solutions of

1) x1 + x2 +x3 + x4= 15
x1 >1, x2 >0, x3 > 1, x4> 2

2) x1 + x2 + x3 + x4 =12
xi >0, i= 1,2,3,4.
• Oct 13th 2009, 10:40 AM
Soroban
Hello, Laura!

Here's the second one . . .the easier of the two.

Quote:

Find the number of integer solutions of: . $\displaystyle x_1 + x_2 + x_3 + x_4 \:=\:12\qquad x_1,x_2,x_3,x_4 \:>\:0$

Consider 12 objects in a row; leave a space between them.

. . $\displaystyle o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o \:\_\:o\:\_\:o\:\_\:o\:\_\:o$

There are 11 spaces.
We choose 3 of them to insert "dividers."

So that: .$\displaystyle o\:\_\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o \:|\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\quad\text { means: }2 + 4 + 1 + 5$

. . And: .$\displaystyle o\:|\:o\:\_\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\: \_\:o\:\_\:o\:|\:o\:\_\:o \quad \text{ means: }1 + 2 + 7 + 2$

Therefore, there are: .$\displaystyle {11\choose3} \:=\:165$ solutions.