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:

\begin{equation*} {19 \choose 7} - \left[{13 \choose 1}{17 \choose 5} - {13\choose 2}{15 \choose 3} + {13\choose 3}{13 \choose 1}\right] = 1716 \text{.} \end{equation*}

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.