The number of non-negative integral solutions of $$x_1~ + ~x_2~ + ~x_3~ + ~...~ + ~x_r~ = ~n$$ can be written as $$^{n+r-1}C_{r-1}$$
This can be found in two ways (as far as I know) -
- Arrange $r-1$ pluses($+$) & $n$ ones($1$) in a row -> $^{n+r-1}C_{r-1}$
For example: $x + y + z = 4$, then $\frac{1}{} \frac{+}{}\frac{1}{}\frac{1}{} \frac{+}{} \frac{1}{}$ is one of the arrangements and we find the total no. of such arrangements
- Finding the coefficient of $x^n$ in $$(x^0 + x^1 + ... + x^n)^r$$
I know that we get the same value [which is $^{n+r-1}C_{r-1}$] from the second method too but why do we find the coefficient of $x^n$ in this series ? What is the understanding behind using the second method ?
I could have said "finding no. of ... is same as finding the coefficient of $n$ in $(1+x)^{n+r-1}$ " but it is not taught like so, which means there is some importance in using that series. So I want to know that importance of the series in finding the no. of arrangements.