Question

The number of positive integral solutions of $x+y+z=n$ is _____ .

Answer 1

Direct formula $\to$

no. of positive integral solution of

$x_1+x_2+......+x_2=n$

${=}^{n+z-1}{C}_{r-1}$

here r=3

${\Rightarrow }^{n+3-1}{C}_{3-1}$

${\Rightarrow }^{n+2}{C}_2$

$\Rightarrow \frac{(n+2)!}{n!2!}=\frac{(n+2)(n+1)(n!)}{n!2}$

$\Rightarrow \frac{(n+1)(n+2)}{2}$