Number of Solutions of a Linear Equation

Number of Solutions of a Linear Equation

## Formula

For non-negative integer solutions of:

$$x_1+x_2+\cdots+x_n=r$$

the number of solutions is:

$${}^{r+n-1}C_{n-1}$$

For positive integer solutions:

$${}^{r-1}C_{n-1}$$

## Usage

- Used to solve counting problems by interpreting variables as numbers of objects assigned to different groups.