Final answer:
The smallest value that can be obtained from the expression (int)(Math.random() * 10) + 1 is 1. It results from the lowest possible value generated by Math.random() being above 0.0, multiplied by 10, cast to an integer, and then 1 is added.
Step-by-step explanation:
The expression given, (int)(Math.random() * 10) + 1, is used to generate random numbers in many programming languages, including Java. Here's how the expression works:
1. Math.random() generates a double value that is greater than or equal to 0.0 and less than 1.0.
2. This value is then multiplied by 10, giving us a range of values from 0.0 to just below 10.0.
3. Applying the (int) cast, the resulting number is truncated down to the nearest whole number, which would be any integer from 0 to 9.
4. Finally, adding 1 to this value shifts the range from 1 to 10 (both inclusive).
Therefore, the smallest value that can be obtained from this expression is when Math.random() returns the smallest value in its range, which is just above 0.0. When this is multiplied by 10 and cast to an (int), it becomes 0. Adding 1 to this gives us the smallest possible value, which is 1.