Test's we can use when two combinations are equal

Question

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.

One obvious but invalid solution comes from setting the equality $2r + 4 = r - 7$, which yields $r = -11$. Obviously, combinations cannot have negative numbers. — Toby Mak, May 03 '17 at 12:36
No. The first thing you have to use is to find out what this syntax means and why: https://en.wikipedia.org/wiki/Binomial_theorem — user76568, May 03 '17 at 12:46
Next thing you will use is the "Post Your Answer" button, combined with the fact you are allowed here. ;) — user76568, May 03 '17 at 12:57

score 4 · Accepted Answer · answered May 03 '17 at 12:43

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 $3$.

Test's we can use when two combinations are equal

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