Inclusion-Exclusion Principle. Determine the number of solutions of the equation $x_1+x_2+x_3+x_4+x_5=14$ in positive integers $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ not exceeding $5$.
I know I need to introduce a new variable, $y$, but then when I do that I get a negative... 
$y_1=x_1-5$
$y_2=x_2-5$
$y_3=x_3-5$
$y_4=x_4-5$
$y_5=x_5-5$
my new equation would be $y_1+y_2+y_3+y_4+y_5=-11$

the number of non-negtive integer solutions to X1+X2+......XK=N is equal to the no of way of placing n identical balls into k balled boxes? Can you see that? 
the no of solutions is n+k-1Ck (where C is for combinations)
Now you need to consider the constraints, note your y's defined above are either 0 or negative (we dont want this), so instead its
(5-X1)+(5-X2)+(5-X3)+(5-X4)+(5-X5)=25-14, which boils down to
Y1+Y2+Y3+Y4+Y5=11, here Yi's are non-negative
You should be able to get the answer from here :)
