combinatorics : divide m objects into n boxes

Question

I have a question, I answer it but I don't know my answer is correct or not. My question is : Twelve teachers are to be assigned to three schools. In how many ways can this be done if i. a specific school (School 1) receives at least 2 teachers? ii. each school receives exactly 4 teachers? My solution is : i. if we get teachers and schools distinct objects and boxes respectively, we can divide 12 teachers into 3 schools in 3^{12} ways. We want the school 1 to have at least 2 teachers. The complement of it is that the school1 has 1 or 0 teacher. If the school1 has 0 teacher then we should divide 12 teachers into 2 schools so we have 2^12 ways. If the school1 has 1 teacher then we should choose 1 teacher of 12 and divide 11 teachers into 2 shools. So, in that case we have 12*2^11. Therefore the total ways are 3^12-(2^12+12*2^11). ii. First we choose 4 of 12 then 4 of 8 and 4 of 4 finally. Hence, the total ways are 485+70+1=556 Is my answer correct? Thanks a lot