Maximize Probability on a Weighted Coin

Daniel has a weighted coin that flips heads 25\frac{2}{5} of the time and tails 35\frac{3}{5} of the time. If he flips it 99 times, the probability that it will show heads exactly nn times is greater than or equal to the probability that it will show heads exactly kk times, for all k=0,1,,9,knk=0, 1,\dots, 9, k\ne n.

If the probability that the coin will show heads exactly nn times in 99 flips is pq\frac{p}{q} for positive coprime integers pp and qq, then find the last three digits of pp.

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