Four dice are rolled. The number of possible outcomes in which atleast one die show 6 is

Options

A. 671
B. 168
C. 176
D. 650

Answer

To solve the problem of finding the number of possible outcomes when four dice are rolled, with the condition that at least one die shows a 6, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Total Outcomes:**
When rolling four dice, each die has 6 faces. Therefore, the total number of outcomes when rolling four dice can be calculated as:
\[ \text{Total Outcomes} = 6^4 \]
\[ 6^4 = 1296 \]

2. **Calculate the Outcomes with No 6s:**
Next, we need to find the number of outcomes where none of the dice show a 6. If a die cannot show a 6, it can only show the numbers 1, 2, 3, 4, or 5. Thus, each die has 5 options. The total number of outcomes with no 6s is:
\[ \text{Outcomes with No 6s} = 5^4 \]
\[ 5^4 = 625 \]

3. **Calculate the Outcomes with At Least One 6:**
To find the number of outcomes where at least one die shows a 6, we subtract the number of outcomes with no 6s from the total outcomes:
\[ \text{Outcomes with At Least One 6} = \text{Total Outcomes} - \text{Outcomes with No 6s} \]
\[ = 1296 - 625 = 671 \]

### Final Answer:
The number of possible outcomes in which at least one die shows a 6 is **671**.

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