Combi offer! #3

There is a very basic method...put x=0,1,2.....8 and solve each equation using beggars or rather stars and bars method(Same method, different names)....the sum turns out to be 1+4+7....25, which is 117

t = 3x can assume values in the set {0, 3, 6, 9, ... , 24}; y and z each in the set 0(1)24. Thus one can represent the possibilities for t by the generating function,

$1+{x}^{3}+{x}^{6}+{x}^{9}+{x}^{12}+{x}^{15}+{x}^{18}+{x}^{21}+{x}^{24}$

and those for x and y by

$1+{x}^{1}+{x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6}+{x}^{7}+{x}^{8}+{x}^{9}+{x}^{10}+{x}^{11}+{x}^{12}+{x}^{13}+{x}^{14}+{x}^{15}+{x}^{16}+{x}^{17}+{x}^{18}+{x}^{19}+{x}^{20}+{x}^{21}+{x}^{22}+{x}^{23}+{x}^{24}$

Multiply these three polynomials together and select the coefficient for ${x}^{24}$ . I used Python sympy for this purpose.