Twelve different books are split into three piles of 3, 4 and 5 books. In how many ways can this be done?

Pick 3 for the small pile: ${}^{12}{C}_3=220$; 4 from the remaining 9: ${}^9{C}_4=126$; the last 5 form the third pile. $220\times 126=27720$. The piles have different sizes, so there is no extra division.