In how many ways we can color $15$ eggs with colors red, blue, and green, when each egg must be colored with exactly two distinct colors.
My answer is :
(1) red and blue colored eggs are in the first box, and
(2) red and green colored eggs are in the second box, and
(3) blue and green colored eggs are in the third box.
So we have $3$ boxes. Any one or two of these boxes can be empty, because we can put all eggs in one box (that is color every egg with same combination of colors, so all are in one box).
Boxes are labeled and objects (eggs) are not.
So the answer is $$n+k-1 \choose k-1 $$ so $$ 15 + 3 -1 \choose 15-1$$ Is it correct?