An ice cream vendor offers three different flavors. A person buys an ice cream with 5 scoops. How many possibilities are there (the order of the scoops does not matter).
A choice of ice creams is three integers $(x, y, z)=$ (chocolate, raspberry, vanilla) with $x+y+z=5$ for example $(2,2,1)$. Let's say the ice cream scoops are in a line. Since the order does not matter, assume the chocolate scoops are on the left, the raspberry scoops in the middle, and the vanilla scoops on the right. Add two separators between the scoops of different flavors. We now have 5 ice cream scoops and 2 separators. The position of the separators gives the number of scoops of each flavor. For example, a separator in position 3 and in position 6 means 2 chocolate scoops (positions 1 and 2), 2 raspberry scoops (positions 4 and 5), and 1 vanilla scoop (position 7). This gives an explicit construction of the different possible combinations of ice creams. We have 7 objects (5 scoops + 2 separators) and 2 separators to place, and thus $\binom{7}{2}$ possibilities.