Solution:
To solve this problem, we can use the concept of combinations.
Given that the bag contains 3 black, 4 white, and 2 red balls, we need to find the number of selections of at most 6 balls containing balls of all the colors.

1. Selection of balls:
Let's consider the selection of balls in the following way:
- Selecting 0 black balls, 0 white balls, and 0 red balls
- Selecting 1 black ball, 0 white balls, and 0 red balls
- Selecting 0 black balls, 1 white ball, and 0 red balls
- Selecting 0 black balls, 0 white balls, and 1 red ball
- Selecting 1 black ball, 1 white ball, and 0 red balls
- Selecting 1 black ball, 0 white balls, and 1 red ball
- Selecting 0 black balls, 1 white ball, and 1 red ball

2. Selections with 0 black balls:
If we select 0 black balls, we have only one option, which is to select 6 white or red balls. Since we have 4 white balls and 2 red balls, the number of ways to select 6 balls is given by:
C(6, 6) + C(6, 6) = 1 + 1 = 2

3. Selections with 1 black ball:
If we select 1 black ball, we have 3 options for selecting the black ball and 5 options for selecting the remaining balls (either white or red). The number of ways to select these balls is given by:
C(3, 1) * C(5, 5) + C(3, 1) * C(5, 5) = 3 + 3 = 6

4. Selections with 0 white balls:
If we select 0 white balls, we have only one option, which is to select 6 black or red balls. Since we have 3 black balls and 2 red balls, the number of ways to select 6 balls is given by:
C(6, 6) + C(6, 6) = 1 + 1 = 2

5. Selections with 1 white ball:
If we select 1 white ball, we have 4 options for selecting the white ball and 5 options for selecting the remaining balls (either black or red). The number of ways to select these balls is given by:
C(4, 1) * C(5, 5) + C(4, 1) * C(5, 5) = 4 + 4 = 8

6. Selections with 0 red balls:
If we select 0 red balls, we have only one option, which is to select 6 black or white balls. Since we have 3 black balls and 4 white balls, the number of ways to select 6 balls is given by:
C(6, 6) + C(6, 6) = 1 + 1 = 2

7. Selections with 1 red ball:
If we select 1 red ball, we have 2 options for selecting the red ball and 5 options for selecting the remaining balls (either black or white). The