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$120, 210 ,3003$ 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?

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See here. 3003 appears eight times, infinitely many numbers appears six times and it is not known whether any number appears more than eight times. –  martini Apr 13 '12 at 6:42
    
fascinating ... –  scibuff Apr 13 '12 at 11:17
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