$120, 210 ,3003$ appear $6$ times in Pascal's triangle.
$120={10\choose3}={16\choose2}={120\choose1}\\$
$210={10\choose4}={21\choose2}={210\choose1}\\$
$3003={14\choose6}={78\choose2}={3003\choose1}$
Are there any numbers $>1$ that appear more than $6$ times, and are there finitely many appearing $6$ times?
