# How often can a number occur in Pascals Triangle?

Each number occurs at least twice as $a={a\choose1}={a\choose a-1}$. If a number occurs somewhere else in the triangle (most likely twice, if it's not of the form ${2a\choose a}$) then that number occurs $4$ times. After that, it becomes interesting. I found the following with a simple script:

$$120={120\choose1}={16\choose 2}={10\choose3}={10\choose7}={16\choose 14}={120\choose119}$$

$$210={210\choose1}={21\choose 2}={10\choose4}={10\choose6}={21\choose 19}={210\choose209}$$

$$1540={1540\choose1}={56\choose 2}={22\choose3}={22\choose19}={56\choose 54}={1540\choose1539}$$

$$7140={7140\choose1}={120\choose 2}={36\choose3}={36\choose33}={120\choose 118}={7140\choose7139}$$

And a special one, that did not only occur six times, but eight:

$$3003={3003\choose1}={78\choose 2}={15\choose5}={14\choose6}={14\choose8}={15\choose10}={78\choose 76}={3003\choose3002}$$

I only checked the numbers up to $10000$; so here's my question

Besides $1$, are there other numbers that occur infinitely often? Is there an upper bound known to how many times a number can occur in Pascal's Triangle?

• Searching on OEIS with just 120, 210, 1540 from your own data, already gives some references. Apr 2, 2017 at 22:21
• Very related: On a unique(?) binomial property of $3003$ Apr 2, 2017 at 23:06
• "Besides 11, are there other numbers that occur infinitely often?" - no, and you can confirm this by noting that after row $n$ ($n>1$), the number $n$ cannot appear. Apr 3, 2017 at 2:36
• "Each number occurs at least twice" - not when $1=a-1$! Apr 3, 2017 at 7:30
• Related MO post: Singmaster's conjecture. I found the link in answers to this question: Are there surprisingly identical binomial coefficients? Apr 3, 2017 at 16:29

It is clear that the only number that appears infinitely many times in Pascal's triangle is $1$, because any other number $x$ can appear only within the first $x + 1$ rows of the triangle.