An ordinary dice is rolled 4 times, numbers appearing on them are listed. The number of different throws, such that the largest number appearing on them is NOT 4 , is :

Options

A. A) 175
B. B) 625
C. C) 1040
D. D) 1121

Answer

To solve the problem of finding the number of different throws when an ordinary die is rolled 4 times such that the largest number appearing on them is NOT 4, we can follow these steps:

### Step 1: Calculate Total Outcomes
When a die is rolled 4 times, each roll can result in one of 6 outcomes (1, 2, 3, 4, 5, or 6). Therefore, the total number of outcomes when rolling the die 4 times is given by:
\[ \text{Total outcomes} = 6^4 \]
Calculating this gives:
\[ 6^4 = 1296 \]

### Step 2: Calculate Outcomes Where the Largest Number is 4
Next, we need to find the number of outcomes where the largest number is exactly 4. This means that the numbers rolled can only be 1, 2, 3, or 4.

The total outcomes when rolling the die 4 times with the numbers limited to 1, 2, 3, and 4 is:
\[ \text{Outcomes with largest number 4} = 4^4 \]
Calculating this gives:
\[ 4^4 = 256 \]

### Step 3: Subtract Outcomes Where the Largest Number is Less Than 4
However, we need to exclude the cases where the largest number is less than 4 (i.e., only 1, 2, or 3 are rolled). The total outcomes with numbers limited to 1, 2, and 3 is:
\[ \text{Outcomes with largest number less than 4} = 3^4 \]
Calculating this gives:
\[ 3^4 = 81 \]

Thus, the number of outcomes where the largest number is exactly 4 can be calculated by:
\[ \text{Outcomes with largest number 4} = 4^4 - 3^4 = 256 - 81 = 175 \]

### Step 4: Calculate Outcomes Where the Largest Number is NOT 4
Now, we can find the number of outcomes where the largest number is NOT 4 by subtracting the outcomes where the largest number is 4 from the total outcomes:
\[ \text{Outcomes where largest number is NOT 4} = \text{Total outcomes} - \text{Outcomes with largest number 4} \]
Substituting the values we calculated:
\[ \text{Outcomes where largest number is NOT 4} = 1296 - 175 = 1121 \]

### Final Answer
Thus, the number of different throws such that the largest number appearing on them is NOT 4 is:
\[ \boxed{1121} \]