Out of 10 white, 8 black and 6 red balls, the number of ways in which one or more balls can be selected is (assuming balls of the same colour are identical)

Options

A. 681
B. 691
C. 679
D. 692

Answer

To solve the problem of selecting one or more balls from a collection of 10 white, 8 black, and 6 red balls (where balls of the same color are identical), we can use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the Number of Choices for Each Color**:
- For white balls: We can select from 0 to 10 white balls. This gives us 11 options (0, 1, 2, ..., 10).
- For black balls: We can select from 0 to 8 black balls. This gives us 9 options (0, 1, 2, ..., 8).
- For red balls: We can select from 0 to 6 red balls. This gives us 7 options (0, 1, 2, ..., 6).

2. **Calculate the Total Number of Combinations**:
- The total number of ways to select balls (including the option of selecting none) is the product of the number of choices for each color:
\[ \text{Total combinations} = (11) \times (9) \times (7) \]

3. **Perform the Multiplication**:
- Calculate \( 11 \times 9 = 99 \)
- Then calculate \( 99 \times 7 = 693 \)

4. **Exclude the Case of Selecting No Balls**:
- Since the problem specifies that we need to select one or more balls, we need to subtract the case where no balls are selected (which is counted in our total combinations):
\[ \text{Valid combinations} = 693 - 1 = 692 \]

5. **Final Answer**:
- Therefore, the number of ways in which one or more balls can be selected is **692**.