how many different choices exist? If we choose $k$ objects from $n$ with replacement and we don't ignore the order of the choices(e.g if we choose $3$ objects of $A,B$ with replacement,the results $AAB$ and $ABA$ are considered as different ones),how many different choices exist?
Is it $\binom{n+k-1}{k}$ ?

Hint: We are forming $k$-letter "words."
The first letter can be chosen in $n$ ways. For each such choice, the second letter can be chosen in $n$ ways. Continue. 
Remark: If we ignore the order, then we are in a "Stars and Bars" situation. Let $y_i$ be the number of Type $i$ objects we select ($i=1$ to $n$). We are looking for the number of solutions of the equation $y_1+y_2+\cdots +y_n=k$ in non-negative integers. In that case, the expression of the OP is correct. However, when $AAB$ and $ABA$ are considered different, the Stars and Bars analysis does not apply. 
