Without any restriction, there would be \({19\choose 7}\) ways to distribute the stars. Now we must use PIE to eliminate all distributions in which one or more student gets more than one star:
Interestingly enough, this number is also the value of \({13 \choose 7}\text{,}\) which makes sense: if each student can have at most one star, we must just pick the 7 out of 13 students to receive them.