Which one of the following numbers can be removed from the set S = 0, 2, 4, 5, 9 without changing the average of set S?

Answer: Option C

The average of the elements in the original set S is:
$$\begin{array}{rl}& \frac{0+2+4+5+9}{5}\\ & =\frac{20}{5}\\ & =4\end{array}$$
If we remove an element that equals the average, then the average of the new set will remain unchanged. The new set after removing 4 is {0, 2, 5, 9}.
The average of the elements is,
$$\begin{array}{rl}& \frac{0+2+5+9}{4}\\ & =\frac{16}{4}\\ & =4\end{array}$$