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Prove that $\gcd({n \choose i},{n \choose j})>1,~0<i,j<n$

My work:

I tried expanding $n \choose i$ and $n \choose j$ to find that there is some number that divides both and after division the numbers are still integers, but I could not prove that they are integers. Please help.

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*Hint:* Suppose that their greatest common divisor is $1$. Then the fraction $$\frac{{n\choose j}}{{n\choose i}}$$ is written in lowest terms.

On the other hand, this fraction can also be written as $$\frac{{n-i\choose n-j}}{{j\choose i}}\ldots$$

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