AMC 8 2017 Problem 20

An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? $\textbf{(A) }\frac{14}{75} \textbf{(B) }\frac{56}{225} \textbf{(C) }\frac{107}{400} \textbf{(D) }\frac{7}{25} \textbf{(E) }\frac{9}{25}$