The number of one-one functions f: \ a, b, c, d\ → \ 0, 1, 2, , 10\...

Given $2f\left(a\right) + 3f\left(c\right) = f\left(b\right) - f\left(d\right)$

Using fundamental principle of counting,

when $f$\left(c\right) = 0, f$\left(a\right) = 1\text{then}f\left(b\right) - f\left(d\right) = 2$ has $7$ choices,

Similarly when $f\left(c\right) = 0, f\left(a\right) = 2\text{then}f\left(b\right) - f\left(d\right)$ has $5$ choices.

Counting like this we get the total number of one-one function as $31$