Consider the equation $$x_1+x_2+...+x_n=m$$ where $x_i$ denotes the number of balls thrown in urn $i$. Now our question minds about cases where at least one $x_i$ is greater or equal to $c$. This is equivalent to find number of the answers of that equation where $x_1\ge c$ or the number of answers of the following equation$$(x_1-c)+x_2+...+x_n=m-c$$which is $$\binom{m+n+1-c}{n+1}$$using stars and bars approach

Hint:

Consider the equation $$x_1+x_2+...+x_n=m$$ where $x_i$ denotes the number of balls thrown in urn $i$. Now our question minds about cases where at least one $x_i$ is greater or equal to $c$. This is equivalent to find number of the answers of that equation where $x_1\ge c$ or the number of answers of the following equation$$(x_1-c)+x_2+...+x_n=m-c$$