Pulling balls from a box

Question

This is a homework problem I just need checked before I hand it over. It seems deceptively easy so I'm not sure if I'm missing something.

In a box there are $10$ balls, each coloured differently. In how many ways can we pull out three balls if after pulling a ball from the box we return it to the box?

We can look at the box as a multiset consisting of ten balls, each of infinite multiplicity(because we return the ball to the box after we pull it out), so the number of ways we can pull out three balls is equal to $10^3$.

Answer 1

take choosing each ball as an independent event seeing as you return the ball after picking it

then the number of possibilities to pick the first ball is 10, as is the second ball and the third

so the total number of possibilities is $10 \times 10 \times 10$ = $10^3$

this is equivalent to $10\choose1$ $\times $ $10\choose1$ $\times $ $10\choose1$ = $10^3$

Answer 2

If you are referring to the number of possible sequences of draws, then yes, it will be $10^3$, but if you want to count the number of possible resulting multi-sets, then it will be the number of ways to put three identical balls into 10 distinct boxes, which is the number of non-negative integer solutions to $x_1+x_2+\cdots+x_{10}=3$, which is ${9+3}\choose{3}$ (to arrange 9 pluses and 3 ones).