The number of ways of distributing 4 identical physics books and  5 identical mathematics books among three students such that each student gets at least one book is  _____.

Total number of ways of distributing 4 identical physics books and  5 identical mathematics books among three students is$\left({ }^{4+3-1}{C}_{3-1}\right)\left({ }^{5+3-1}{C}_{3-1}\right){=}^6{C}_2\left({ }^7{C}_2\right)=15\left(21\right)=315$

A= A does not get any book, B= b does not get any book, C= C does not get any book

n(A) =Number of ways of distributing the books to B,C

$\begin{array}{l}n\left(A\right)={}^{4+2-1}C_{2-1}\times {}^{5+2-1}C_{2-1}=30\\ n\left(B\right)=n\left(C\right)=30\\ n(A\cap B)={}^{4+1-1}C_{1-1}{}^{5+1-1}C_{1-1}=1=(B\cap C)=n(C\cap A)\\ n(A\cap B\cap C)=0\end{array}$

Required no. of ways $=315-(30+30+30-1-1-1+0)=228$