A die is thrown four times and the sum of the numbers is noted. If the sum of the numbers is 23 then the probability is

Options

A. (a) `1/324`
B. (b) `1/342`
C. (c) `1/243`
D. (d) `1/322`

Answer

To solve the problem of finding the probability of getting a sum of 23 when a die is thrown four times, we can follow these steps:

### Step 1: Determine the Total Number of Possible Outcomes
When a die is thrown once, there are 6 possible outcomes (1, 2, 3, 4, 5, or 6). Since the die is thrown 4 times, the total number of possible outcomes is calculated as follows:

\[ \text{Total Outcomes} = 6^4 = 1296 \]

### Step 2: Find the Favorable Outcomes for a Sum of 23
Next, we need to find all the combinations of numbers on the die that can sum up to 23 when the die is thrown four times. The maximum sum we can get from four throws is 24 (if all throws result in 6). Therefore, the only combinations that yield a sum of 23 are:

1. (6, 6, 6, 5)
2. (6, 6, 5, 6)
3. (6, 5, 6, 6)
4. (5, 6, 6, 6)

These combinations can be arranged in different orders, but they all yield the same sum of 23.

### Step 3: Count the Favorable Outcomes
From the combinations identified, we see that there are 4 distinct arrangements that give us the sum of 23.

\[ \text{Number of Favorable Outcomes} = 4 \]

### Step 4: Calculate the Probability
The probability of an event is given by the formula:

\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]

Substituting the values we found:

\[ \text{Probability} = \frac{4}{1296} \]

### Step 5: Simplify the Probability
Now, we simplify the fraction:

\[ \frac{4}{1296} = \frac{1}{324} \]

### Conclusion
Thus, the probability of getting a sum of 23 when a die is thrown four times is:

\[ \text{Probability} = \frac{1}{324} \]

### Final Answer
The correct answer is option A: \( \frac{1}{324} \).

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