Question

Find the number of integral solution to |x|+|y|+|z| = 15.

Answer 1

The total number of solutions will be equal to 850 

Answer 2

The formula is 4N^2 + 2. (where N is the sum in the equation of the question. Therefore, it's 4* (15^2) + 2 =902.

Don't know how to derive the formula though. Anyway, I had come up with my own way, counting solutions where --

all numbers where +ve,

one in (x,y,z) was -ve,

two in (x,y,z) was -ve.

all numbers (three nos.) were -ve

upon summation of these the answer was 902 as well.

Remember one -ve (x,y,z) would entail multiplication by 3 as e.g.(-4,5,6) is different from (4,-5,6) and (4,5,-6) [3 solutions]. Likewise, (-4,-5,6) is diff from (-4,5,-6) and (4,-5,-6).