5. Put 10 identical balls into 3 boxes numbered $1, 2, 3$ (all 10 balls must be placed each time), with the requirement that the number of balls in each box is no less than the box's number. The number of such ways to place the balls is (   .
(A) 9
(B) 12
(C) 15
(D) 18
5.C.

Let the number of balls in the boxes numbered $1, 2, 3$ be $x_{1}, x_{2}, x_{3}$ respectively, then
$$
\begin{array}{l}
x_{1}+x_{2}+x_{3}=10\left(x_{1} \geqslant 1, x_{2} \geqslant 2, x_{3} \geqslant 3\right) . \\
\text { Let } y_{1}=x_{1}, y_{2}=x_{2}-1, y_{3}=x_{3}-2, \text { then } \\
y_{1}+y_{2}+y_{3}=7\left(y_{1} \geqslant 1, y_{2} \geqslant 1, y_{3} \geqslant 1\right) .
\end{array}
$$

Thus, the problem is transformed into finding the number of positive integer solutions to the equation $y_{1}+y_{2}+y_{3}=7$. It is easy to know that the number of solutions is $\mathrm{C}_{5}^{2}=15$.