Number Theory but

Well the only soln. Is x=y=z=o

We can simply use the Well Ordering Principle or The method of Infinite Descent taking the advantage of our given domain. Or else, the option for inequality is also there since our domain is positive. This question can be solved in a lot many methods!! :)

We can rewrite the equation as $\large\ { x }^{ 3 }+{ 5y }^{ 3 }+25{ z }^{ 3 }=15xyz$

Now, let us apply $AM-GM$ inequality to the left hand side of the equation

i.e., $\large\ { x }^{ 3 }+5{ y }^{ 3 }+25{ z }^{ 3 }\ge \quad 3\sqrt [ 3 ]{ { x }^{ 3 }{ .y }^{ 3 }.{ z }^{ 3 }.5.25 }$

i.e., $\large\ { x }^{ 3 }+5{ y }^{ 3 }+25{ z }^{ 3 }\ge \quad 3.5xyz$

i.e., $\large\ { x }^{ 3 }+5{ y }^{ 3 }+25{ z }^{ 3 }\ge \quad 15xyz$

and the equality occurring iff $\large\ { x }^{ 3 }=5{ y }^{ 3 }=25{ z }^{ 3 }$

And by inspection , the only possible solution is $x=y=z=0$ .

Hence , no. of solution = $\boxed{1}$