One more java random number question

Question

which of the following codes are correct to generate a random number between start and end inclusively. Both methods seem to be working for me but im pretty sure one would be incorrect
int num = (int)(start + (Math.random() * (end-start)));
int num = (int)(start+1 + (Math.random() * (end-start)));

Answer 1

I say neither of them work. Let's use 5 and 8 as our start and end. In the Math class, random() is said to generate a random number, x, where 0 <= x < 1.
The first one:
First, multiply by 8 - 5 (3).
0 <= x < 3
Then, add 5.
5 <= x < 8
Then, casting to an int, we get a number 5 - 7.
The second one:
First, multiply by 8 - 5 (3).
0 <= x < 3
Then, add 6.
6 <= x < 9
Then, casting to an int, we get a number 6 - 8.
The "correct" way:
int num = (int) (Math.random() * (end - start + 1) + start);First, multiply by 8 - 5 + 1 (4).
0 <= x < 4
Then, add 5.
5 <= x < 9
Then, casting to an int, we get a number 5 - 8.