The number of four-digit numbers that can be formed by using the digits `1,2,3,4,5,6,7,8` and `9` such that the least digit used is `4`, when repetition of digits is allowed is

Options

A. `617`
B. `671`
C. `716`
D. `761`

Explanation

To solve the problem of finding the number of four-digit numbers that can be formed using the digits `1, 2, 3, 4, 5, 6, 7, 8, 9` such that the least digit used is `4`, and repetition of digits is allowed, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Identify the digits available**: Since the least digit used is `4`, the possible digits we can use are `4, 5, 6, 7, 8, 9`. This gives us a total of 6 digits to work with.

2. **Case Analysis**: We will analyze different cases based on how many times the digit `4` appears in the four-digit number.

**Case 1**: `4` appears exactly once.
- We can choose 1 position out of 4 for the digit `4`. The number of ways to choose 1 position from 4 is given by \( \binom{4}{1} = 4 \).
- The remaining 3 positions can be filled with any of the 6 digits (including `4`), so there are \( 6^3 = 216 \) ways to fill these positions.
- Total for Case 1: \( 4 \times 216 = 864 \).

**Case 2**: `4` appears exactly twice.
- We can choose 2 positions out of 4 for the digit `4`. The number of ways to choose 2 positions from 4 is given by \( \binom{4}{2} = 6 \).
- The remaining 2 positions can be filled with any of the 6 digits, so there are \( 6^2 = 36 \) ways to fill these positions.
- Total for Case 2: \( 6 \times 36 = 216 \).

**Case 3**: `4` appears exactly three times.
- We can choose 3 positions out of 4 for the digit `4`. The number of ways to choose 3 positions from 4 is given by \( \binom{4}{3} = 4 \).
- The remaining 1 position can be filled with any of the 6 digits, so there are \( 6^1 = 6 \) ways to fill this position.
- Total for Case 3: \( 4 \times 6 = 24 \).

**Case 4**: `4` appears in all four positions.
- In this case, the only number we can form is `4444`.
- Total for Case 4: \( 1 \).

3. **Summing Up All Cases**: Now, we add the totals from all cases to find the total number of four-digit numbers that can be formed.
\[ \text{Total} = 864 + 216 + 24 + 1 = 1105 \]

### Final Answer:
The total number of four-digit numbers that can be formed is **1105**.