If the number of five digit numbers with distinct digit and 2 at the `10^(th)` place is 336 k, then k is equal to :

Options

A. 8
B. 7
C. 4
D. 6

Answer

To solve the problem of finding the value of \( k \) in the number of five-digit numbers with distinct digits where 2 is fixed at the \( 10^{th} \) place, we can follow these steps:

### Step 1: Understand the structure of the five-digit number
A five-digit number can be represented as \( ABCDE \), where \( B \) (the \( 10^{th} \) place) is fixed as 2. Thus, our number looks like \( A2CDE \).

### Step 2: Determine the possible digits for \( A, C, D, \) and \( E \)
The digits available for selection are \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \). Since 2 is already used in the \( 10^{th} \) place, we cannot use it again. Additionally, \( A \) cannot be 0, as it would not be a five-digit number.

### Step 3: Choose the first digit \( A \)
The first digit \( A \) can be any digit from the set \( \{1, 3, 4, 5, 6, 7, 8, 9\} \) (8 options, since 0 and 2 are excluded).

### Step 4: Choose the remaining digits \( C, D, \) and \( E \)
After selecting \( A \), we have 8 remaining digits (including 0 but excluding the digit chosen for \( A \) and 2). We need to choose 3 more digits from these remaining 8 digits.

### Step 5: Calculate the number of ways to choose and arrange the digits
1. Choose \( A \): 8 choices.
2. Choose 3 digits from the remaining 8 digits: \( \binom{8}{3} \).
3. The chosen 3 digits can be arranged in \( 3! \) ways.

### Step 6: Combine the counts
The total number of five-digit numbers can be calculated as:
\[ \text{Total} = 8 \times \binom{8}{3} \times 3! \]

### Step 7: Calculate \( \binom{8}{3} \) and \( 3! \)
- \( \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \)
- \( 3! = 6 \)

### Step 8: Substitute back into the total count
\[ \text{Total} = 8 \times 56 \times 6 = 2688 \]

### Step 9: Relate to the given equation
We are given that this total equals \( 336k \):
\[ 2688 = 336k \]

### Step 10: Solve for \( k \)
\[ k = \frac{2688}{336} = 8 \]

Thus, the value of \( k \) is \( 8 \).