5. Place 10 identical balls into three 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 arrangements 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 $x_{1}+x_{2}+x_{3}=10$ (where $x_{1} \geqslant 1, x_{2} \geqslant 2, x_{3} \geqslant 3$).
Let $y_{1}=x_{1}, y_{2}=x_{2}-1, y_{3}=x_{3}-2$, then $y_{1}+y_{2}+y_{3}=7$ (where $y_{1} \geqslant 1, y_{2} \geqslant 1, y_{3} \geqslant 1$).
The problem is then transformed into finding the number of positive integer solutions to the equation $y_{1}+y_{2}+y_{3}=7$, which is easily known to be $C_{6}^{2}=15$.