Copies of Trump’s official paperwork with the draft board are open to all at the website called Smoking Gun. He signed up with his local board 4 days after his 18th birthday. He apparently took the four one-year student deferments to go through college but then reported to the draft board. Something was picked up during his physical so he was asked to return in 4 months which he did. He signed up then and was put in the draft pool and very luckily for him and others with higher numbers, his turned out to be 356 out of the top of 365.
Only if the war had expanded would he would have been called but he did go through the entire process and didn’t take advantage of things by say going to get a Masters which would have bought him more deferments. That was a common practice as was getting a PhD.
Actually they ended student deferments when they switched to the lottery system. I was a 19 year old student and drew a high number also.