Randomly sample table cells - equal N across rows and columns

I plan to have 12 people answer 300 questions. Each subject will answer 100 questions, and each question answered by 4 subjects.

For various reasons, the assignment must be random. Here is how I approach this but am open to any ideas.

I created a blank 300*12 data frame (300 rows named by question ids and 12 columns for subjects). For each subject column, randomly sample 100 rows and put "1" in the 100 cells. As a result, I can make sure each subject is assigned to 100 questions randomly, but not all questions get answered by exactly 4 subjects.

Please advise.