Question

Find the number of different ways in which $8$ different books can be distributed among $3$ students, if each student receives atleast $2$ books.

Answer 1

Case 1: They can be distributed 2,2 and 4.

We choose the student to get 4 books in 3 ways .The 4 books to give to that students in ${}^8{C}_4$ ways.

The 4 books can be distributed among 2 students by ${}^{\$}{C}_2$ ways.The other students automatically gets the remaining 2 books in 1 way = 3${\ast }^8{C}_4{\ast }^4{C}_2\ast 1$=1260 ways.

Case 2: They can be distincted 2,3 and 3. We choose the students to get 2 books in 3 way.

2 books can be given to that student in ${}^4{C}_2$ ways.

Remaining 6 books can be distributed among 3 students in ${}^6{C}_3$ ways.

The other student automatically gets the remaining books in 1 ways

$\therefore$ $3{\ast }^8{C}_2{\ast }^6{C}_3\ast 1$=1680 ways

$\therefore$ Total =1260+1680=2940 ways.