Four Equations in Four Unknowns

The first and last choices can be dropped since the four values do not add to 10. Knowing the formulas $\displaystyle\sum_{k=1}^{n}n=\frac{n(n+1)}{2}$ and $\displaystyle\sum_{k=1}^{n}n^3=\left(\frac{n(n+1)}{2}\right)^2$ helps since the "sum of the cubes of the first $n$ integers" is the square of the "sum of the integers". This is true for equations 1 and 3 in the problem ( $100=10^2$ ), so the solution is probably $1,2,3,4$ in any order. Also $4!=24$ and the sum of the squares works out to be 30. Thus the solution is definitely $\boxed{1,2,3,4}$ in any order.

$1 + 2 + 3 + 4 = 10$ is the only decomposition of 10 with 4 distinct digits. Then you just have to check the other equations. That's precisely the reason why the answer is {1, 2, 3, 4} and not (1, 2, 3, 4) because the order doesn't matter.