Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $k,m,n\in \mathbb{N}$ where $1 < k < n-1$. Consider the equation $$\binom{n}{k} = m!$$ which can also be equivalently written as $$n!=(n-k)!k!m!$$ The only instances I found are $\binom{4}{2} = 3!$ and $\binom{10}{3} = 5!$ I do not see any pattern coming out. As I went far out, it seemed that it is hard to find other examples as the second instance seems to be related to the problem of consecutive numbers being composed of only small primes. Is it true that these are the only instances?


share|cite|improve this question
@Henry, thanks for the edit. – Tibebu Haile Yilma Mar 20 '11 at 18:00
There is also ${16 \choose 2}=5!$. – Henry Mar 20 '11 at 18:01
@Henry, are there more that do not include small numbers such as 2? – Tibebu Haile Yilma Mar 20 '11 at 18:05
I just wrote a script and I don't believe there are any others up to 13! (except of course $\binom{n}{0}$ and $\binom{n}{1}$). – Ben Alpert Mar 20 '11 at 18:12
up vote 9 down vote accepted

You're asking about the existence of factorials in entries of Pascal's triangle. I googled for factorials pascal's triangle and found this discussion:

share|cite|improve this answer
Thanks very much. I constructed the Pascal triangle to some extent while looking for examples but did not consider googling in those terms. – Tibebu Haile Yilma Mar 21 '11 at 14:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.