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 follow these steps:

### Step 1: Understand the selection process
We have three types of balls:
- White balls: 10
- Black balls: 8
- Red balls: 6

Since the balls of the same color are identical, we can use the concept of combinations to find the number of ways to select balls.

### Step 2: Calculate the number of ways to select each color
For each color of balls, we can select from 0 to the maximum number of balls available of that color. Therefore, the number of ways to select balls of each color is given by:
- For white balls: We can select 0 to 10 balls, which gives us \(10 + 1 = 11\) options (including selecting none).
- For black balls: We can select 0 to 8 balls, which gives us \(8 + 1 = 9\) options (including selecting none).
- For red balls: We can select 0 to 6 balls, which gives us \(6 + 1 = 7\) options (including selecting none).

### Step 3: Calculate the total combinations
The total number of ways to select the balls (including the option of selecting none) is the product of the options for each color:
\[ \text{Total ways} = (10 + 1)(8 + 1)(6 + 1) = 11 \times 9 \times 7 \]

### Step 4: Calculate the product
Now we calculate the product:
\[ 11 \times 9 = 99 \]
\[ 99 \times 7 = 693 \]

### Step 5: Exclude the case of selecting no balls
Since we need to find the number of ways to select **one or more** balls, we must subtract the case where no balls are selected (which is 1 way):
\[ \text{Ways to select one or more balls} = 693 - 1 = 692 \]

### Final Answer
Thus, the total number of ways to select one or more balls is **692**.

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