Numbers Appearing 8 Times in Pascal's Triangle

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Theorem

Excluding $1$, the number $3003$ is the smallest integer to appear $8$ times in Pascal's triangle.

No other number below $2^{23}$ appears as often.


Proof

\(\ds 3003\) \(=\) \(\, \ds \frac {3003!} {3002! \times 1!} \, \) \(\, \ds = \, \) \(\ds \dbinom {3003} 1\)
\(\ds \) \(=\) \(\, \ds \frac {78!} {76! \times 2!} \, \) \(\, \ds = \, \) \(\ds \dbinom {78} 2\)
\(\ds \) \(=\) \(\, \ds \frac {15!} {10! \times 5!} \, \) \(\, \ds = \, \) \(\ds \dbinom {15} 5\)
\(\ds \) \(=\) \(\, \ds \frac {14!} {8! \times 6!} \, \) \(\, \ds = \, \) \(\ds \dbinom {14} 6\)
\(\ds \) \(=\) \(\, \ds \frac {14!} {6! \times 8!} \, \) \(\, \ds = \, \) \(\ds \dbinom {14} 8\)
\(\ds \) \(=\) \(\, \ds \frac {15!} {5! \times 10!} \, \) \(\, \ds = \, \) \(\ds \dbinom {15} {10}\)
\(\ds \) \(=\) \(\, \ds \frac {78!} {2! \times 76!} \, \) \(\, \ds = \, \) \(\ds \dbinom {78} {76}\)
\(\ds \) \(=\) \(\, \ds \frac {3003!} {1! \times 3002!} \, \) \(\, \ds = \, \) \(\ds \dbinom {3003} {3002}\)


This theorem requires a proof.
In particular: It remains to be shown that there are no other occurrences of $3003$, and that there are no other numbers less than $2^{23}$ with this property.
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Also see


Sources

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