Test's we can use when two combinations are equal If we have the expression:$$\binom{n}{2r+4}=\binom{n}{r-7}$$ then what are some formulas we can use to get to the value of $r$ straightaway? I has to use trial and error after expanding the expression, and i know formulas related to such combinations exist but somehow i couldn't find them on the internet. Any help will be appreciated.

For binomial coefficients we have the equivalence:

$$\binom{n}{k} = \binom{n}{l} \Leftrightarrow (k=l \lor k = n-l)$$

This can be seen from the definition of the binomial coefficient, which makes the LHS identity equivalent to the denominator being the same, that is $(n-k)!k! = (n-l)!l!$.

So your example becomes that either

$$2r+4 = r-7$$ $$r = -11$$

which is non-allowed, or

$$2r+4 = n - r + 7$$ $$r = n/3 + 1$$

Which is allowed if $n$ is a multiple of $3$.