The whole numbers less than 51 are {0, 1, 2, ...50}, which makes 51 of them.
The sum of the first 50 consecutive positive integers, 1+2+3+4+...+50 is given by the Gauss formula for addition:
[tex] \frac{50*51}{2}=25*51= 1275[/tex]
To maximize "(sum of 50 different whole numbers, each less than 51)-(sum of 45 different whole numbers, each less than 51)":
we need to maximize the first part, and minimize what we are subtracting.
That is:
we need to maximize "sum of 50 different whole numbers, each less than 51" which is done by subtracting the smallest one, that is 0, from 1275
and minimize "sum of 45 different whole numbers, each less than 51"
which is done by subtracting the greatest 6 numbers from 1275, that is:
1275-50-49-48-47-46-45=990
Thus, the difference is at most: 1275-990=285
Remark: we need to subtract 6 numbers because there are in total 51, and we are adding the smallest 45 of them.
Answer: D: 285
