1989 AIME Problem 5

When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to 0^{}_{} and is the same as that of getting heads exactly twice. Let \frac ij^{}_{}, in lowest terms, be the probability that the coin comes up heads in exactly 3_{}^{} out of 5^{}_{} flips. Find i+j^{}_{}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.