The number of different numbers of six digits each can be formed from the digits 4 5 6 7 8 9 such that they are not divisible by 5 is

Options

A. 720
B. 600
C. 500
D. 400

Answer

To solve the problem of finding the number of different six-digit numbers that can be formed from the digits 4, 5, 6, 7, 8, and 9 such that they are not divisible by 5, we can follow these steps:

### Step 1: Determine the Total Number of Six-Digit Combinations
Since we have 6 unique digits (4, 5, 6, 7, 8, 9), the total number of different six-digit numbers that can be formed using all these digits is given by the factorial of the number of digits.

\[ \text{Total combinations} = 6! = 720 \]

### Step 2: Identify the Condition for Divisibility by 5
A number is divisible by 5 if it ends in either 0 or 5. Since our available digits are 4, 5, 6, 7, 8, and 9, the only digit that can make a number divisible by 5 is 5. Therefore, we need to calculate how many six-digit numbers can be formed that end with the digit 5.

### Step 3: Calculate the Number of Combinations Ending with 5
If the last digit is fixed as 5, we have 5 remaining digits (4, 6, 7, 8, 9) to arrange in the first five positions. The number of ways to arrange these 5 digits is given by the factorial of the number of digits.

\[ \text{Combinations ending with 5} = 5! = 120 \]

### Step 4: Calculate the Number of Combinations Not Divisible by 5
To find the number of six-digit numbers that are not divisible by 5, we subtract the number of combinations that are divisible by 5 from the total combinations.

\[ \text{Combinations not divisible by 5} = \text{Total combinations} - \text{Combinations ending with 5} \]
\[ \text{Combinations not divisible by 5} = 720 - 120 = 600 \]

### Final Answer
The number of different six-digit numbers that can be formed from the digits 4, 5, 6, 7, 8, and 9 that are not divisible by 5 is **600**.

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