TITLE: Are there surprisingly identical binomial coefficients?
QUESTION [9 upvotes]: Suppose $\binom{n}{k}=\binom{n'}{k'}$ with $k \geq 2$, $k' \geq 2$, $n \geq 2k$ and $n' \geq 2k'$. Does it follow that $n=n'$ and $k=k'$? 
EDIT: Yup, $\binom{16}{2}=\binom{10}{3}=120$. 
Now I want to ask if there are infinitely many such pairs, but I should probably ask that in a separate question. Thanks!
EDIT 2: for future reference, yes there are infinitely many such coincidences. See Singmaster's Conjecture.

REPLY [7 votes]: $$
\binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6}.
$$
Nobody knows whether any number besides 3003 appears at least eight times in Pascal's triangle.