Find the number of 7 -digit numbers which can be formed by using all digits 6,5,4,5,3,4,3 only once such that the odd digits occupy odd places only ?

Options

A. 24
B. 15
C. 18
D. 20

Answer

To solve the problem of finding the number of 7-digit numbers that can be formed using the digits 6, 5, 4, 5, 3, 4, 3, with the condition that the odd digits occupy the odd places only, we can follow these steps:

### Step 1: Identify the Digits
The digits we have are:
- Odd digits: 5, 5, 3, 3
- Even digits: 6, 4, 4

### Step 2: Determine the Positions
In a 7-digit number, the positions are:
- Odd positions: 1, 3, 5, 7 (4 positions)
- Even positions: 2, 4, 6 (3 positions)

### Step 3: Arrange the Odd Digits
We need to place the odd digits (5, 5, 3, 3) in the odd positions.

The number of ways to arrange these digits is calculated using the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2!} \]
Where:
- \( n \) is the total number of items to arrange (4 in our case),
- \( n_1 \) is the count of the first repeated item (5 appears 2 times),
- \( n_2 \) is the count of the second repeated item (3 appears 2 times).

Thus, we have:
\[ \text{Arrangements of odd digits} = \frac{4!}{2! \cdot 2!} = \frac{24}{2 \cdot 2} = 6 \]

### Step 4: Arrange the Even Digits
Next, we need to place the even digits (6, 4, 4) in the even positions.

Using the same formula:
\[ \text{Arrangements of even digits} = \frac{3!}{2!} \]
Where:
- \( n = 3 \) (total even digits),
- \( n_1 = 2 \) (4 appears 2 times).

Thus:
\[ \text{Arrangements of even digits} = \frac{6}{2} = 3 \]

### Step 5: Calculate the Total Arrangements
To find the total number of 7-digit numbers, we multiply the number of arrangements of odd digits by the number of arrangements of even digits:
\[ \text{Total arrangements} = \text{Arrangements of odd digits} \times \text{Arrangements of even digits} = 6 \times 3 = 18 \]

### Final Answer
Therefore, the total number of 7-digit numbers that can be formed under the given conditions is **18**.

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