Algebra problem #1126

Every positive integer $n$ occurs in Pascal's triangle as the entries $\binom{n}{1}$ and $\binom{n}{n-1}$ . The only time this does not give two distinct entries is when $1 = n -1$ , so $n = 2$ . For any row higher than the second row, all entries are either $1$ or larger than $2$ , so $2$ only occurs once, while every other positive integer occurs at least twice. So the answer is $1$ .