Maximize Probability on a Weighted Coin

Daniel has a weighted coin that flips heads $\frac{2}{5}$ of the time and tails $\frac{3}{5}$ of the time. If he flips it $9$ times, the probability that it will show heads exactly $n$ times is greater than or equal to the probability that it will show heads exactly $k$ times, for all $k=0, 1,\dots, 9, k\ne n$.

If the probability that the coin will show heads exactly $n$ times in $9$ flips is $\frac{p}{q}$ for positive coprime integers $p$ and $q$, then find the last three digits of $p$.