The total number of ways of selecting 10 balls out of an unlimited number of identical white, red and blue balls is equal to

Options

A. `""^(12)C_(2)`
B. `"^(12)C_(3)`
C. `""^(10)C_(2)`
D. `""^(10)C_(3)`

Answer

To solve the problem of selecting 10 balls from an unlimited supply of identical white, red, and blue balls, we can use the "stars and bars" combinatorial method. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We need to select 10 balls from three different colors: white, red, and blue. Since the balls are identical within each color, we can represent the selection of balls as a problem of distributing indistinguishable objects (balls) into distinguishable boxes (colors).

### Step 2: Set Up the Equation
Let:
- \( x_1 \) = number of white balls selected
- \( x_2 \) = number of red balls selected
- \( x_3 \) = number of blue balls selected

We need to satisfy the equation:
\[ x_1 + x_2 + x_3 = 10 \]
where \( x_1, x_2, x_3 \geq 0 \).

### Step 3: Apply the Stars and Bars Theorem
The stars and bars theorem states that the number of ways to distribute \( n \) indistinguishable objects into \( r \) distinguishable boxes is given by:
\[ \binom{n + r - 1}{r - 1} \]

In our case:
- \( n = 10 \) (the total number of balls)
- \( r = 3 \) (the number of colors)

### Step 4: Substitute Values into the Formula
Using the formula:
\[ \text{Number of ways} = \binom{10 + 3 - 1}{3 - 1} = \binom{12}{2} \]

### Step 5: Calculate the Binomial Coefficient
Now we calculate \( \binom{12}{2} \):
\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

### Final Answer
Thus, the total number of ways of selecting 10 balls out of an unlimited number of identical white, red, and blue balls is **66**.

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