$\begingroup$

The object of the board game "The Settlers of Catan" is to obtain 10 "victory points". There are five ways to obtain victory points:

Settlements, worth 1 victory point. Each player starts the game with 2 settlements and can build up to 3 more, for a total of 5. Cities, worth 2 victory points. Cities are built on top of settlements, replacing them in doing so. Players can build up to 4 cities. Victory Point development cards, worth 1 victory point. Players can buy up to 5 such cards from the deck. The Longest Road card, either held or not held, worth 2 victory points. The Largest Army card, either held or not held, worth 2 victory points.

I remember that we can decompose this into the number of integer solutions to the equation:

$$ s + 2c + v + 2r + 2a = 10 $$ where $$ 0\leq s \leq 5; 0 \leq c \leq 4; 0 \leq v \leq 5; 0 \leq r \leq 1; 0 \leq a \leq 1 $$

and

$$ s + c \geq 2 $$

to account for the 2 starting settlements that can be upgraded into cities.

Also, it is possible to have 9 victory points and then take Longest Road or Largest Army, bringing your total to 11. I think the number of additional solutions here is

$$ \text{Solutions}(s + 2c + v = 9) + 2\times\text{Solutions}(s + 2c + v = 7) $$

multiplying by 2 to account for interchangeability of Longest Road and Largest Army, with bounds as above.

However, I don't remember how to find the number of solutions to these equations.

At commenters' request, here are some example solutions: