Let ' m ' denotes the number of four digit numbers such that the left most digit is odd, the second digit is even and all four digits are different and ' n ' denotes the number of four digit numbers such that left most digit is even, second digit is odd and all four digit are different. If m=nk, then k equals :

Options

A. `(4)/(5)`
B. `(3)/(4)`
C. `(5)/(4)`
D. `(4)/(3)`

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) based on the given conditions and then determine the value of \( k \) such that \( m = n \cdot k \).

### Step 1: Calculate \( m \)

1. **Choosing the first digit (odd)**: The first digit must be odd. The odd digits available are 1, 3, 5, 7, and 9. Thus, there are 5 choices for the first digit.

\[ \text{Choices for 1st digit} = 5 \]

2. **Choosing the second digit (even)**: The second digit must be even. The even digits available are 0, 2, 4, 6, and 8, giving us 5 choices. However, since the first digit is already chosen and it is odd, we can choose any of the 5 even digits.

\[ \text{Choices for 2nd digit} = 5 \]

3. **Choosing the third digit**: The third digit can be any digit that is not already chosen. We have already used 1 odd digit and 1 even digit, so we have 8 digits left (10 total digits - 2 used).

\[ \text{Choices for 3rd digit} = 8 \]

4. **Choosing the fourth digit**: The fourth digit must also be different from the previous three digits, leaving us with 7 choices.

\[ \text{Choices for 4th digit} = 7 \]

5. **Calculating \( m \)**: Now, we multiply the number of choices together to find \( m \).

\[ m = 5 \times 5 \times 8 \times 7 = 1400 \]

### Step 2: Calculate \( n \)

1. **Choosing the first digit (even)**: The first digit must be even. The available even digits are 2, 4, 6, and 8 (0 cannot be the first digit). Thus, there are 4 choices for the first digit.

\[ \text{Choices for 1st digit} = 4 \]

2. **Choosing the second digit (odd)**: The second digit must be odd. The odd digits available are 1, 3, 5, 7, and 9, giving us 5 choices.

\[ \text{Choices for 2nd digit} = 5 \]

3. **Choosing the third digit**: The third digit can be any digit that is not already chosen. We have already used 1 even digit and 1 odd digit, so we have 8 digits left.

\[ \text{Choices for 3rd digit} = 8 \]

4. **Choosing the fourth digit**: The fourth digit must also be different from the previous three digits, leaving us with 7 choices.

\[ \text{Choices for 4th digit} = 7 \]

5. **Calculating \( n \)**: Now, we multiply the number of choices together to find \( n \).

\[ n = 4 \times 5 \times 8 \times 7 = 1120 \]

### Step 3: Find \( k \)

We know that \( m = n \cdot k \). Substituting the values we found:

\[ 1400 = 1120 \cdot k \]

To find \( k \):

\[ k = \frac{1400}{1120} = \frac{5}{4} \]

### Final Answer

Thus, the value of \( k \) is:

\[ \boxed{\frac{5}{4}} \]