This is a result of Singmaster (American Math Monthly, 1971).

 Here is his proof ($N(a)$ is the number of times $a$ appears.) This result has since been much improved, see the [wikipedia article on Singmaster's conjecture.][1]



[![enter image description here][3]][3]

To elaborate somewhat, Abbott-Erdos-Hanson show:
[![enter image description here][2]][2]
and use this to show the better asymptotic bound. It is not quite clear how to use this to get better bounds for all $t.$

  [1]: https://www.wikiwand.com/en/Singmaster%27s_conjecture
  [2]: https://i.sstatic.net/QAuRx.png
  [3]: https://i.sstatic.net/MrceL.png