I see how 落 can be a failure. But how is 第 relevant to failing an exam, is
this a meaning other than 'number, -th' or implicitly falling _below first_
place?
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etymology
What is the origin of 落第?
According to [this website about etymologies](http://gogen-
allguide.com/ra/rakudai.html),

> 「第」は「第何位」などと用いられるように、一段一段の段階をや物事の順序を意味し、転じて、段階ごとの試験も意味するようになった。
>
> そこから、落第は段階ごとの試験に落ちる意味で「不合格」を表すようになった。
>
> 中国では「不合格」の意味で用いられるが、日本では「落第生」と言うように、不合格になることで進級や卒業ができないことも意味する。

Paraphrasing the relevant part about 第: the 第 comes from 第何位 and referred to
the stairs in a staircase or the ordering of things. It then evolved to refer
to tests that comes in tiers/steps. Then, 落第 would mean to fall down from a
step in the sequence of tests.
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I suggest that「落第」is a Chinese loanword. It originally meant _failure to be
selected as a ranked Imperial bureaucrat due to unsatisfactory results from
the[Imperial
Examinations](https://en.wikipedia.org/wiki/Imperial_examination)_.

> [元](https://en.wikipedia.org/wiki/Yuan_dynasty) 曾瑞 《留鞋記》楔子：＂人都道我落第無顏，羞歸鄉里＂
>
> _Everyone spoke about me not being selected for Imperial service, and to
> ashamedly return to my home village_

  * 「落」means _to fall to the ground_ > _fail_
  * 「第」refers to the _grade-ranking result_ of a candidate's attempt at the Imperial Examinations (「科第」in Chinese)

> [北齊](https://en.wikipedia.org/wiki/Northern_Qi) 顏之推 《顏氏家訓・勉學》：＂明經求第，則顧人答策。＂
>
> _[Testing] one's thorough knowledge of the classics and to obtain Imperial
> rank, candidates are invited to answer these questions._

* * *

**Reference:**

  * 《{{kr:漢語}}大詞典》
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