To solve the problem of determining how many different five-digit lock codes are possible under the given conditions, we can follow these steps:
### Step 1: Identify the digits for the first position
The first digit must be odd. The available odd digits are 1, 3, 5, 7, and 9. Therefore, there are 5 possible choices for the first digit.
**Hint:** Remember that the first digit must be an odd number, and list all odd digits available.
### Step 2: Identify the digits for the last position
The last digit must also be odd, but it cannot be the same as the first digit (since repetition is not allowed). Since one odd digit has already been used in the first position, there are 4 remaining choices for the last digit.
**Hint:** Consider that you have already used one odd digit for the first position, reducing the options for the last position.
### Step 3: Identify the digits for the second position
The second digit can be any digit from 0 to 9, but it cannot be the same as the first or last digits. Since two digits have already been used, there are 10 total digits minus 2 used digits, leaving us with 8 choices for the second digit.
**Hint:** Count all possible digits (0-9) and subtract the digits already used.
### Step 4: Identify the digits for the third position
Similar to the second position, the third digit can also be any digit from 0 to 9, but it cannot be the same as the first, last, or second digits. Therefore, there are 10 total digits minus 3 used digits, which gives us 7 choices for the third digit.
**Hint:** Again, start with the total digits and subtract those already chosen.
### Step 5: Identify the digits for the fourth position
For the fourth digit, it can also be any digit from 0 to 9, but it cannot be the same as the first, last, second, or third digits. Thus, there are 10 total digits minus 4 used digits, resulting in 6 choices for the fourth digit.
**Hint:** Keep track of how many digits have been used to determine the options left.
### Step 6: Calculate the total number of lock codes
Now that we have the number of choices for each position, we can calculate the total number of different lock codes by multiplying the number of choices for each position together:
\[
\text{Total Codes} = (\text{Choices for 1st}) \times (\text{Choices for 2nd}) \times (\text{Choices for 3rd}) \times (\text{Choices for 4th}) \times (\text{Choices for 5th})
\]
Substituting the values we found:
\[
\text{Total Codes} = 5 \times 8 \times 7 \times 6 \times 4
\]
Calculating this gives:
\[
5 \times 8 = 40
\]
\[
40 \times 7 = 280
\]
\[
280 \times 6 = 1680
\]
\[
1680 \times 4 = 6720
\]
Thus, the total number of different lock codes possible is **6720**.
### Final Answer
The total number of different lock codes possible is **6720**.
