There are three piles of identical red, blue and green balls and each pile contains at least 10 balls. The number of ways of selecting 10 balls if twice as many red balls as green balls are to be selected, is

Options

A. 3
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to determine the number of ways to select 10 balls from three piles of identical red, blue, and green balls, with the condition that twice as many red balls as green balls are selected. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( x \) = number of green balls selected
- \( 2x \) = number of red balls selected (since we need twice as many red balls as green)
- \( y \) = number of blue balls selected

### Step 2: Set Up the Equation
The total number of balls selected must equal 10:
\[ x + 2x + y = 10 \]
This simplifies to:
\[ 3x + y = 10 \]

### Step 3: Express \( y \) in terms of \( x \)
From the equation \( 3x + y = 10 \), we can express \( y \) as:
\[ y = 10 - 3x \]

### Step 4: Determine Valid Values for \( x \)
Since we are selecting balls, \( x \) must be a non-negative integer. Additionally, \( y \) must also be non-negative. Therefore, we need:
\[ 10 - 3x \geq 0 \]
This leads to:
\[ 3x \leq 10 \implies x \leq \frac{10}{3} \implies x \leq 3 \]
Thus, the possible values for \( x \) are \( 0, 1, 2, \) and \( 3 \).

### Step 5: Calculate Corresponding Values of \( y \)
Now we can find the corresponding values of \( y \) for each valid \( x \):
- If \( x = 0 \):
\[ y = 10 - 3(0) = 10 \]
- If \( x = 1 \):
\[ y = 10 - 3(1) = 7 \]
- If \( x = 2 \):
\[ y = 10 - 3(2) = 4 \]
- If \( x = 3 \):
\[ y = 10 - 3(3) = 1 \]

### Step 6: Count the Combinations
Each combination of \( (x, y) \) gives us a valid selection of balls:
1. \( (0, 10) \) → 0 green, 0 red, 10 blue
2. \( (1, 7) \) → 1 green, 2 red, 7 blue
3. \( (2, 4) \) → 2 green, 4 red, 4 blue
4. \( (3, 1) \) → 3 green, 6 red, 1 blue

### Step 7: Total Number of Ways
We have 4 valid combinations of \( (x, y) \) that satisfy the conditions of the problem.

Thus, the total number of ways to select the balls is **4**.

### Final Answer
The number of ways of selecting 10 balls is **4**.

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