Estimate number of songs a radio station has

Question

Imagine the following problem:

You listen to a radio station and take notes how often was each song played. How can you estimate based on your notes (e.g. 30 songs played once, 2 played twice, one song three times) how many songs has the radio station available when we assume all the songs are played with a uniform distribution?

Answer 1

You could determine the probability that the observed frequency of songs occurs for each number $N$ of songs and maximize it.