5. There are 10 identical red balls, 15 identical black balls, and 20 identical white balls. Distribute all these balls to two boys and one girl, with each boy receiving at least two balls of each color, and each girl receiving at least three balls of each color. How many different ways are there to distribute the balls to these three students?
5.47250.

When each boy receives two balls of each color and each girl receives three balls of each color, 2 × 2 + 3 = 7 balls of each color have been distributed. Thus, there remain 10 - 7 = 3 red balls, 15 - 7 = 8 black balls, and 20 - 7 = 13 white balls.

The 3 remaining red balls can be distributed among the three students in $\mathrm{C}_{5}^{2}=10$ ways, the 8 remaining black balls can be distributed among the three students in $\mathrm{C}_{10}^{2}=45$ ways, and the 13 remaining white balls can be distributed among the three students in $\mathrm{C}_{15}^{2}=105$ ways. Therefore, there are $10 \times 45 \times 105 = 47250$ different ways to distribute all the balls to two boys and one girl as required.