Find the number of ways of distributing 20 chocolates, 20 gummy bears, and 20 lemon drops among two people, so that each person gets 30 candies. (The candies of each type are indistinguishable.)
We can write a computer program:
for a in range(20)
for b in range(20)
for c in range(20)
if a + b + c = 30 then ++count
output: count = 298
+118608 Find the number of ways of distributing 20 chocolates, 20 gummy bears, and 20 lemon drops among two people, so that each person gets 30 candies. (The candies of each type are indistinguishable.)
If the computer program is properly written and that is the answer then I accept that.
However I am going to attempt if for myself.
Let one of the halves be
x chocolates, y gummy bears and 30-x-y lemon drops
\(0\le x \le20\\ 0\le y\le20\\ 0\le 30-x-y \le20 \)
the number of dots in the whole square is 21*21 = 441
The number of dots in the triangles is 2(1+2+3+...10) = 2(55)=110
441-110 = 331 combinations ..... maybe
I think there is some double counting here.
Lets see.
What I have done below is pinpointed all the dots that have been double counted.
The point in the middle has been counted four times.
The lines are
x=y
x=30-x-y
and
y=30-x-y
There are 11 points on each of the 3 lines which makes 33 overcounts
331-33= 298 combinations altogether. This is the same answer that guest got with his/her program. 
