Coefficients in binomial expansions of $(x+y)^n$

Question

Is it true that if we expand $(x+y)^n$ where $n$ is a prime number, then all the coefficients are divisible (except the first and last term) by $n$?

Note. There are many examples of when $n$ isn't prime, and the coefficients are not divisible by $n$.

Write out the definition of the general term pCr where p is the prime, and notice that it is an integer, but the p in the numerator has not cancelled out — Shailesh, Sep 18 '15 at 14:12
this is known as Freshman's dream and was already asked quite often, just check out the link or use the search - you'll find what you're looking for for sure, like here — user190080, Sep 18 '15 at 14:13

score 3 · Answer 1 · answered Sep 18 '15 at 14:10

3

Hint:

$$(x+y)^n = \sum_{i=0}^n{n\choose i} x^iy^{n-i}$$

So all you need to prove is that ${p\choose i}$ is divisible by $p$ if $p$ is prime.

answered Sep 18 '15 at 14:10

5xum

1 Answers1