hi, joojei, I think our chinese friends explained the point very well, But perhaps you have still doubts, so I will try to explain it in another manner: 学校 is school, but 校 alone means also school. When you have read 在校学生, you thought perhaps that 校学生 was a single word, and you read: 在 + 校学生。But, in fact, you should have read: 在校 + 学生, which literally means: in the school" + "student". In the school works here as an adjective. So, a possible translation to your sentence could be: "A 21-year old student of the school". Hope it helps.

學校 means school, but the phrase 校學 does not mean anything. 校 by itself means school, unless if you mean 小學 or 小學生 which mean elementary school/elementary school student, respectively. Providing a sentence for context purposes would be more helpful.

I think by "校学” you probably mean “小学” (xiao3xue2)，meaning primary school or elementary school, 小学生 means elementary school student。

學校： school 校学： it's the name what we call school in ancient China. but we dont use now. Here are a sentence telling where the "校学“ from? 夏曰校, 殷曰序, 周曰庠,学则三代共之,皆所以明人伦也。——《孟子》 We use different words like "校”“序”庠“学” to express school in different periods time.

在校学生 should be interpreted as 在 校 学生, which is an abbreviation of 在 学校 的 学生. In many short phrases, the preposition 的 is simply omitted. And 校 is an abbreviation of 学校. In Mandarin Chinese, there are many cases where people prefer to use abbreviations for collocations. I can hardly give any clue on how these abbreviations are formed. It is just a matter of memorization.