How many ways are there to distribute 5 balls into 3 boxes, under additional conditions?

Solution 1:

1) Your answer is correct; for each ball, you can choose any box, and every choice is distinguishable at any time.

2) You want to distribute your 5 distinguishable balls into 3 indistinguishable boxes. Let $B(5,3)$ denote the number of ways in which this can be done into exactly 3 indistinguishable non-empty boxes, and use the recurrence relation $B(n,k)=B(n-1,k-1)+kB(n-1,k)$ with $B(n,1)=1$ and $B(n,n)=1$. You seek $B(5,1)+B(5,2)+B(5,3)$. For further reference, see Stirling number.

3) You want to separate your 5 indistinguishable balls into 3 distinguishable boxes. Note that every admissible separation uniquely corresponds to a permutation of the string $XXbbbbb$.

4) You want to partition your 5 indistinguishable balls into 3 indistinguishable boxes. Let $p(5,3)$ denote the number of ways in which this can be done into exactly 3 indistinguishable non-empty boxes, and use the recurrence relation $p(n,k)=p(n-1,k-1)+p(n-k,k)$ with $p(n,1)=1$ and $p(n,n)=1$. You seek $p(5,1)+p(5,2)+p(5,3)$. For further reference, see partition.