How many ways to give five different books to three students?
A student can have all of them.
My attempt
1) Three ways to give separately five each (5,0,0) - 3 ways
2) 4 gifts for one and one gift for another - $^5C_4\times3\times2$
3) 3,1,1 - $^5C_3\times3\times^2C_1\times^1C_1$
4) 2,2,1 - $^5C_2\times3\times^3C_2\times^1C_1$
On
Each way of distributing the $5$ different books corresponds to assigning each book to one of the students we number by $1,2,3$.
So, you can see the number of ways as the number of sequences of length $5$ consisting of the digits $1,2,3$.
For example $11113$ corresponds to the distribution where book $1$ through $4$ go to student $1$ and book $5$ goes to student $3$.
The number of such sequences is obviously $3^5$.
The first textbook can go to any of the three children. Hence each textbook has three choices of whom to go to. This gives us number of permutations as $N=3\times3\times3\times3\times3=3^5$