permutations problem - student may not win all 23 prizes

Problem is:

3 prizes, one for English, one for French and one for Spanish can be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if:

(i) no student may win > 1 prize.

(ii) no student may win all 3 prizes.

(i) I can do ok - 20 * 19 * 18 = 6840. ie 20 students can win E, leaving 19 to win F, etc.

But I am struggling with (ii). If the question asked was student CAN win 1,2 or 3 prizes then poss ways would be 20*20*20=8000. Is that correct.

My first thought was that way to do this is 20*20*19 - but that is incorrect. Correct way to solve is apparently 20*20*20 - 20. But why?

Angus