Total number of 6-digit numbers in which only and all the five digit 1,2,5,7 and 9 appear, is :

Options

A. 56
B. 6!
C. `1/2(6!)`
D. `5/2(6!)`

Answer

To find the total number of 6-digit numbers that can be formed using the digits 1, 2, 5, 7, and 9, where one of these digits is repeated, we can follow these steps:

### Step 1: Understand the Problem
We need to create a 6-digit number using the digits 1, 2, 5, 7, and 9. Since we only have 5 unique digits, one of these digits must be repeated to make a total of 6 digits.

### Step 2: Choose the Digit to Repeat
We can choose any one of the 5 digits (1, 2, 5, 7, or 9) to repeat. The number of ways to choose 1 digit from 5 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, it is \( \binom{5}{1} \).

\[ \text{Ways to choose the digit to repeat} = \binom{5}{1} = 5 \]

### Step 3: Arrange the Digits
Now we have 6 digits in total: 5 unique digits and 1 repeated digit. The arrangement of these digits can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p!} \]

Where:
- \( n \) is the total number of digits (6 in this case),
- \( p \) is the number of times the repeated digit appears (2 in this case).

So, the total arrangements will be:

\[ \text{Total arrangements} = \frac{6!}{2!} \]

Calculating \( 6! \) and \( 2! \):

\[ 6! = 720 \quad \text{and} \quad 2! = 2 \]

Thus,

\[ \text{Total arrangements} = \frac{720}{2} = 360 \]

### Step 4: Combine the Choices and Arrangements
Now, we multiply the number of ways to choose the digit to repeat by the total arrangements:

\[ \text{Total 6-digit numbers} = \text{Ways to choose the digit} \times \text{Total arrangements} \]

\[ \text{Total 6-digit numbers} = 5 \times 360 = 1800 \]

### Final Answer
The total number of 6-digit numbers in which only and all the five digits (1, 2, 5, 7, and 9) appear is **1800**.

---