Answer
See below.
Work Step by Step
We know that probability$=\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}.$
The number of good outcomes is $1$.
If we want to choose $k$ elements out of $n$ regarding the order, not allowing repetition, we can do this in $_{n}P_k=\frac{n!}{(n-k)!}$ ways.
The order matters here when choosing the five balls, thus we have to use permutations. We have $10$ balls for $5$ spots, thus the number of all outcomes is: $_{10}P_{5}=\frac{10!}{(10-5)!}=30240$
Thus probability=$\frac{1}{30240}$