In a single throw with four dice, the probability of throwing seven is

Options

A. `4/6^(4)`
B. `8/6^(4)`
C. `16/6^(4)`
D. `20/6^(4)`

Answer

To find the probability of throwing a sum of 7 with four dice, we can follow these steps:

### Step 1: Calculate the Total Outcomes
When throwing one die, there are 6 possible outcomes (1 through 6). Therefore, when throwing four dice, the total number of outcomes is given by:
\[ \text{Total Outcomes} = 6^4 \]

### Step 2: Identify Favorable Outcomes
Next, we need to find the combinations of the numbers on the four dice that add up to 7. We can list the combinations:

1. **Combination: (1, 1, 1, 4)**
- The number of ways to arrange this combination is calculated using the formula for permutations of multiset:
\[ \text{Ways} = \frac{4!}{3! \cdot 1!} = 4 \]

2. **Combination: (1, 1, 2, 3)**
- The number of ways to arrange this combination is:
\[ \text{Ways} = \frac{4!}{2! \cdot 1! \cdot 1!} = 12 \]

3. **Combination: (1, 2, 2, 2)**
- The number of ways to arrange this combination is:
\[ \text{Ways} = \frac{4!}{1! \cdot 3!} = 4 \]

### Step 3: Calculate Total Favorable Outcomes
Now, we add the number of ways from each combination to find the total number of favorable outcomes:
\[ \text{Total Favorable Outcomes} = 4 + 12 + 4 = 20 \]

### Step 4: Calculate the Probability
Finally, the probability of throwing a sum of 7 with four dice is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{20}{6^4} \]

### Final Answer
Thus, the probability of throwing a sum of 7 with four dice is:
\[ \frac{20}{6^4} \]