Suppose a lock has three numbers in the combination. The first number has to be between 0 and 39. The second number has to be between 9 and 27. The third number has to be between 9 and 37. How long would it take to try all possible combinations if each combination takes 5 seconds to try?

To determine how long it would take to try all possible combinations, we need to calculate the total number of combinations and then multiply that by the time taken to try each one.

### Step 1: Total number of combinations

The number of choices for each number in the combination is:
- **First number**: This has to be between 0 and 39, so there are 40 possible choices (0 to 39 inclusive).
- **Second number**: This has to be between 9 and 27, so there are \( 27 - 9 + 1 = 19 \) possible choices (inclusive).
- **Third number**: This has to be between 9 and 37, so there are \( 37 - 9 + 1 = 29 \) possible choices (inclusive).

The total number of combinations is:
\[
40 \times 19 \times 29 = 22,040
\]

### Step 2: Time to try all combinations

Each combination takes 5 seconds, so the total time to try all combinations is:
\[
22,040 \times 5 = 110,200 \text{ seconds}
\]

### Step 3: Convert time into hours

To convert from seconds to hours:
\[
\frac{110,200}{3600} \approx 30.61 \text{ hours}
\]

Thus, it would take approximately **30.61 hours** to try all possible combinations.

Would you like further details or have any questions?

Here are some related questions:
1. How would the total time change if the second number had 10 more choices?
2. What if the time to try each combination was reduced to 3 seconds?
3. How many combinations would there be if the range of the first number changed to 0-49?
4. Can we reduce the number of combinations if there are restrictions on repeating numbers?
5. What happens to the total time if the second and third ranges overlap?

**Tip**: When dealing with large numbers, converting units like seconds to minutes or hours can give a clearer understanding of the scale.

Discover how to calculate the total time to try all combinations of a lock with three numbers where each number has specific ranges. Learn how combinatorics and unit conversion are applied to solve this problem. Suitable for Grades 7-9.

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