Picking $r$ balls from $k$ types of balls where $k-1$ types of balls have infinite supply and the $k^\text{th}$ type has only $1$ copy.

Question

The question is as follows:

We need to pick $r$ balls. The sports shop has k types of balls, with an infinite number of each type of ball $c_1, c_2,\dots, c_{k−1}$ but only $1$ copy of $c_k$. In how many ways can you pick these $r$ balls?

My solution:

There are $2$ possibilities: $C_k$ is chosen or it is not. if chosen then there are still $k$ options for the other $r-1$ balls left to be selected $= (r-1)^k$

if it is not chosen then there are $k-1$ options for each of the $r$ balls to be selected $= r^{k-1}$

Therefore the total number of ways to select the balls $= (r-1)^k + r^{k-1}$

I am not sure if my solution is correct. Alternate proof methods are also welcome.