Solutions of the Equation... (Awesome)

Range of x,y and z

99 >= x >= 1

48 >= y >= 1

32 >= z >= 1

the system contains one equation and 3 variables we have to take two values arbitrary (y and z i have taken)

when y=1

         z can be 1,2,3....32

when y=2

         z can be 1,2,3....31

when y=3

         z can be 1,2,3....31

and so it goes on till

when y=47

         z can be 1 only

when y=48

         z can be 1 only

Solutions=32+31+31+30+29+29+.....+2+1+1

=2(1+3+5+..+31)+(2 + 4+6+...32)

=512+272=784

Hint : You would look for the coefficient of $x^{94}$ in the taylor expansion of $(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}$ about $0$ .

Python solution:

n=0

for x in range(34):

     for y in range(51):

    for z in range(100):

        if x*3+y*2+z == 94:

            n += 1

$\color{#EC7300}{\text{print n}}$

$\color{#3D99F6}{784}$