Let's set one side a rectangle as x and the other distinct one as y.
The perimeter would be x+x+y+y=24, or 2x+2y=24. This can be simplified to x+y=12.
Since the area is xy, we just have to find how many combinations of positive integer numbers add up to 12.
1 and 11 (area of 11)
2 and 10 (area of 20)
3 and 9 (area of 27)
4 and 8 (area of 32)
5 and 7 (area of 35)
6 and 6 (area of 36)
That's it.
So, this rectangle could have 6 distinct areas.
Hope this helps!
