To find the total number of five-digit numbers formed using the digits {1, 2, 3, 4, 5, 6} that are divisible by 55, we need to consider the divisibility rules for both 5 and 11, since 55 = 5 × 11.
### Step-by-Step Solution:
1. **Divisibility by 5**:
- A number is divisible by 5 if its last digit is either 0 or 5. Since we are using the digits {1, 2, 3, 4, 5, 6}, the only option for the last digit (E) is 5.
- Therefore, the five-digit number will be of the form \( ABCD5 \).
2. **Divisibility by 11**:
- For a number to be divisible by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be either 0 or a multiple of 11.
- In our case, the odd-positioned digits are \( A, C, 5 \) and the even-positioned digits are \( B, D \).
- The condition can be expressed as:
\[
(A + C + 5) - (B + D) \equiv 0 \mod 11
\]
3. **Setting up the equation**:
- Rearranging gives:
\[
A + C + 5 - B - D = 0 \quad \text{or} \quad A + C - B - D = -5
\]
- This means that \( A + C - B - D \) must equal -5.
4. **Finding valid combinations**:
- We need to find combinations of \( A, B, C, D \) from the digits {1, 2, 3, 4, 6} (since 5 is already used as the last digit).
- The possible pairs for \( (A, C) \) and \( (B, D) \) need to satisfy the equation \( A + C = B + D - 5 \).
5. **Case Analysis**:
- We can analyze different cases based on the values of \( A + C \) and find corresponding \( B + D \).
**Case 1**: \( A + C = 6 \)
- Possible pairs: (1, 5), (2, 4), (3, 3)
- Corresponding \( B + D = 11 \) (not possible with available digits).
**Case 2**: \( A + C = 7 \)
- Possible pairs: (1, 6), (2, 5), (3, 4)
- Corresponding \( B + D = 12 \) (not possible).
**Case 3**: \( A + C = 8 \)
- Possible pairs: (2, 6), (3, 5), (4, 4)
- Corresponding \( B + D = 13 \) (not possible).
**Case 4**: \( A + C = 9 \)
- Possible pairs: (3, 6), (4, 5)
- Corresponding \( B + D = 14 \) (not possible).
**Case 5**: \( A + C = 10 \)
- Possible pairs: (4, 6)
- Corresponding \( B + D = 15 \) (not possible).
6. **Counting Valid Combinations**:
- After checking all possible combinations, we find that there are no valid combinations of \( A, B, C, D \) that satisfy the divisibility rule for 11 while keeping the last digit as 5.
### Conclusion:
Since no valid combinations satisfy both conditions, the total number of five-digit numbers formed by the digits {1, 2, 3, 4, 5, 6} that are divisible by 55 is **0**.
