Four dice are rolled. The number of possible outcomes in which atleast one die shows 4 is :

Options

A. 671
B. 168
C. 176
D. none of these

Answer

To solve the problem of finding the number of possible outcomes when rolling four dice, where at least one die shows a 4, we can follow these steps:

### Step 1: Calculate the Total Possible Outcomes
When rolling a single die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). Therefore, when rolling 4 dice, the total number of possible outcomes can be calculated using the formula:

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

Calculating this gives:

\[ 6^4 = 6 \times 6 \times 6 \times 6 = 1296 \]

### Step 2: Calculate the Outcomes Where No Die Shows a 4
Next, we need to find the number of outcomes where none of the dice show a 4. If we exclude the number 4, there are 5 possible outcomes for each die (1, 2, 3, 5, 6). Thus, the number of outcomes where no die shows a 4 is given by:

\[ \text{Outcomes with no 4} = 5^4 \]

Calculating this gives:

\[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]

### Step 3: Calculate the Outcomes Where At Least One Die Shows a 4
To find the number of outcomes where at least one die shows a 4, we can subtract the number of outcomes where no die shows a 4 from the total number of outcomes:

\[ \text{Outcomes with at least one 4} = \text{Total Outcomes} - \text{Outcomes with no 4} \]

Substituting the values we calculated:

\[ \text{Outcomes with at least one 4} = 1296 - 625 = 671 \]

### Final Answer
Thus, the number of possible outcomes in which at least one die shows a 4 is:

\[ \boxed{671} \]

---