my textbook says that that no. of solutions to the equation x+y+z=c is given by

the co efficient of x^c in the expansion of {(1+x^2+x^3+...........+x^c)}^3.

but the mechanism is not mentioned here.

i have been able to get that it is just the no. of triplets of digits of nos. less than c that give a sum of c.but how do they connect this concept to the binomial expansion given above.