What is the total number of study schedules for a seven day period for four subjects where each subject is allotted at least one day? A student wants to make up a schedule for a seven-day period during which she will study one subject each day. What is the number of schedules that devote at least one day to each subject? **My attempt at the solution** Since there are $\ 7!/3!$ ways to schedule four subjects for four of the seven days, and $\ 4^3$ ways to schedule three subjects for three remaining days, total number of ways to schedule the subjects are $\ (7!/3!)\times4^3 $. But this is not the correct answer. Please explain what is wrong with this reasoning.

I'm not quite sure if this is correct. So we could have distributions of type $a)$ $$4,1,1,1$$ or b) $$3,2,1,1$$ or c) $$2,2,2,1$$ The number of distribution of type a) is $4\times {7\choose 4}\cdot 3\cdot 2 =840$

The number of distribution of type b) is $12\times {7\choose 3}\cdot {4\choose 2}\cdot{2\choose 1}=5040$

The number of distribution of type c) is $4\times {7\choose 2}\cdot {5\choose 2}\cdot {3\choose 2}=2520$

So we have $840+5040+2520 = 8400$ schedules.