Numbers Appearing 8 Times in Pascal's Triangle

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

Singmaster's Conjecture

Sources

Apr. 1971: David Singmaster: How Often Does an Integer Occur as a Binomial Coefficient? (Amer. Math. Monthly Vol. 78, no. 4: pp. 385 – 386) www.jstor.org/stable/2316907

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3003$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3003$