Three six-faced dice are thrown together. The probability that the sum of the numbers appearing on the dice is `k(9 le k le 14)`, is

Options

A. `(21 k-k^(2)-83)/(216)`
B. `(k^(2)-3k+2)/(432)`
C. `(21 k-k^(2)-83)/(432)`
D. none of these

Explanation

To solve the problem of finding the probability that the sum of the numbers appearing on three six-faced dice is \( k \) (where \( 9 \leq k \leq 14 \)), we will follow these steps:

### Step 1: Determine the Total Outcomes
When three six-faced dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing three dice is:
\[ 6 \times 6 \times 6 = 216 \]

### Step 2: Identify the Favorable Outcomes
Next, we need to find the number of favorable outcomes for each possible value of \( k \) from 9 to 14. The sum of the numbers on the dice can range from 3 (if all dice show 1) to 18 (if all dice show 6). We will calculate the number of combinations that yield sums of 9, 10, 11, 12, 13, and 14.

### Step 3: Use Generating Functions
To find the number of ways to achieve a certain sum, we can use the generating function for a single die:
\[ x + x^2 + x^3 + x^4 + x^5 + x^6 = x \frac{1 - x^6}{1 - x} \]
For three dice, the generating function becomes:
\[ \left( x \frac{1 - x^6}{1 - x} \right)^3 = x^3 (1 - x^6)^3 (1 - x)^{-3} \]
We are interested in the coefficients of \( x^k \) for \( k = 9, 10, 11, 12, 13, 14 \).

### Step 4: Calculate Coefficients
To find the coefficients of \( x^k \) in the expansion, we can use the binomial theorem and polynomial expansion techniques. The coefficients can be calculated as follows:

1. **For \( k = 9 \)**: Calculate the coefficient of \( x^6 \) in \( (1 - x^6)^3 (1 - x)^{-3} \).
2. **For \( k = 10 \)**: Calculate the coefficient of \( x^7 \).
3. **For \( k = 11 \)**: Calculate the coefficient of \( x^8 \).
4. **For \( k = 12 \)**: Calculate the coefficient of \( x^9 \).
5. **For \( k = 13 \)**: Calculate the coefficient of \( x^{10} \).
6. **For \( k = 14 \)**: Calculate the coefficient of \( x^{11} \).

Using combinatorial methods or generating functions, we can find the number of favorable outcomes for each \( k \).

### Step 5: Calculate the Probability
The probability \( P \) that the sum of the numbers on the dice equals \( k \) is given by:
\[ P(k) = \frac{\text{Number of favorable outcomes for } k}{216} \]

### Final Step: Compile Results
After calculating the favorable outcomes for \( k = 9, 10, 11, 12, 13, 14 \), we can summarize the probabilities.