5. Seven balls of different colors are placed into three boxes numbered 1, 2, and 3. It is known that the number of balls in each box is not less than its number. The number of different ways to place the balls is $\qquad$
5.455.
(1) If the number of balls placed in boxes 1, 2, and 3 are 2, 2, and 3 respectively, then the number of different ways to place them is $\mathrm{C}_{7}^{2} \mathrm{C}_{5}^{2}=210$;
(2) If the number of balls placed in boxes 1, 2, and 3 are 1, 3, and 3 respectively, then the number of different ways to place them is $\mathrm{C}_{7}^{1} \mathrm{C}_{6}^{3}=140$;
(3) If the number of balls placed in boxes 1, 2, and 3 are 1, 2, and 4 respectively, then the number of different ways to place them is $\mathrm{C}_{7}^{1} \mathrm{C}_{6}^{2}=105$.
Thus, the total number of ways is $210+140+105=455$.