No. of selections of one or more balls from 10 white, 9 black and 4 red balls, are:

Options

A. 360
B. 359
C. 549
D. None of these

Answer

To solve the problem of finding the number of selections of one or more balls from 10 white, 9 black, and 4 red balls, we can follow these steps:

### Step 1: Calculate the Total Number of Balls
First, we need to determine the total number of balls available for selection. We have:
- 10 white balls
- 9 black balls
- 4 red balls

So, the total number of balls is:
\[ 10 + 9 + 4 = 23 \]

### Step 2: Understand the Selection of Balls
We need to find the number of ways to select one or more balls from these 23 balls. The selections can range from selecting just one ball to selecting all 23 balls.

### Step 3: Use the Binomial Coefficient Formula
The number of ways to select \( k \) balls from \( n \) balls is given by the binomial coefficient \( \binom{n}{k} \). Therefore, the total number of ways to select any number of balls (from 0 to 23) is given by the sum:
\[ \sum_{k=0}^{23} \binom{23}{k} \]
This sum represents all possible selections, including selecting no balls at all.

### Step 4: Apply the Binomial Theorem
According to the binomial theorem, the sum of the binomial coefficients is:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
For our case, where \( n = 23 \):
\[ \sum_{k=0}^{23} \binom{23}{k} = 2^{23} \]

### Step 5: Exclude the Case of Selecting No Balls
Since we are interested in selecting one or more balls, we need to exclude the case where no balls are selected (which corresponds to \( k = 0 \)). The number of ways to select at least one ball is:
\[ 2^{23} - 1 \]

### Final Calculation
Now, we calculate \( 2^{23} \):
\[ 2^{23} = 8388608 \]
Thus, the number of ways to select one or more balls is:
\[ 8388608 - 1 = 8388607 \]

### Conclusion
The total number of selections of one or more balls from the given collection is:
\[ \boxed{8388607} \]