While studying application of Multinomial theorem in PnC I got stuck in two questions :
In how many ways the sum of upper faces of four distinct dice can be six ?
The textbook gave the following solution :
Let $x_1 , x_2 , x_3$ and $x_4$ be the numbers on the upper faces of the cube , then
$x_1 + x_2 + x_3 + x_4 = 6$ , where $1 \leq x_1 , x_2 , x_3 ,x_4 , x_5 , x_6 \leq 6$
No. of ways of getting x = Coefficient of $p^6$ in the expansion $(p+p^2+p^3+p^4+p^5+p^6)^4$
Or coefficient of $p^2$ in the series $(1+p+p^2+p^3+p^4+p^5)$
We can expand the series till infinite terms as the terms with highest power of p are not considered while calculating the coefficient of $p^2$
And then the textbook had an infinite series of G.P. which equations to $(1-p)^{-4}$ . In another question there was almost the same situation of sum of 3 variables equals to a constant
$x_1 + x_2 + x_3 = 10$
But here it had a constraint that $x_1 , x_2 , x_3$ are non negative integers and $x_1$ < 6 , $x_2$ < 7 , $x_3$ < 8. They followed the same procedure of creating series and finding coefficients , but the series was not expanded to infinite terms to which the textbook states
In each bracket the series is not extended to infinite terms because the upper limit of each variable is less than 10.
Why we can't extend it ? Why we can't follow the same procedure as we did we the dice question and say that the upper terms won't count and Why specifically it works if upper limit of each variable is equal to or greater than the mentioned constant on the right hand side ?