At a large high school, there are $20$ senior homerooms each with between $20$ and $25$ students. A random sample of $40$ seniors is to be selected as follows.

Each homeroom teacher will be asked to place the names of the students in their homeroom into a box and then randomly select two names. The two students selected from each of the $20$ homerooms will then be combined to form the sample of size $40$.

Will this method result in a simple random sample of seniors at this high school?