Find the number of one-one functions $f:\{a,b,c,d\}\to\{0,1,2,3,...,10\}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$.

Question

Find the number of one-one functions $f:\{a,b,c,d\}\to\{0,1,2,3,...,10\}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$.

My Attempt

I rearranged the equation like this $f(b)=2f(a)+3f(c)+f(d)$.

Now, if $f(c)=0$ then $2f(a)+f(d)\leq 10$

Can the problem be solved by any method other than counting all cases manually. I kept on taking cases and ended up with 31 cases.

The answer given was $31$. Can it be done by some other approach