Beast ball league ways to get 10 points I know we can start by making the choices +1, +2, and +3. We can find how many ways to get 1 point, 2 points, 3 points, 4 points... But, I can’t find a pattern to find out how many ways you can get 10 points.
Let $B_{k, W}$ be the number of ways to get $k$ points ending with a Win.
Let $B_{k, L}$ be the number of ways to get $k$ points ending with a Lose.      

We have the recurrence relations:   


   $ B_{k, W } = B_{k-3, W } + B_{k-2, L} $
 $ B_{k, L} = B_{k-1, W} + B_{k-1, L}$
 The starting conditions are $B_{1, W} = 0 , B_{1, L } = 1, B_{2,W} = 1, B_{2, L} = 1, B_{3,W} = 1, B_{3,L} = 2$.   


From here, we can find $ B_{10, W} + B_{10, L}$, which is left to the reader.
We are essentially trying to find the number of nonnegative solutions to $x+2y+3z=10$. To do this, consider the $\textbf{generating function}$ $$G(x)=(1+a+a^2+a^3+\cdots)(1+a^2+a^4+a^6+\cdots)(1+a^3+a^6+a^9+\cdots).$$
You are trying to find the coefficient of $x^{10}$ in this expression, and I'll leave for you to do. ^_^