I'm trying to create a tutorial series to accompany a statistical mechanics lecture course. While browsing the material I saw the following binomial coefficient identity.
$$\binom{N}{n+1} = \frac{N-n}{n+1}\binom{N}{n}$$
I've tried to use other proofs as guidelines but for whatever reason I can't seem to prove it.
Thanks in advance.
Notice that: $${N\choose n+1}=\frac{N!}{(n+1)!(N-(n+1))!}=\frac{N-n}{n+1}\frac{N!}{n!(N-n)(N-(n+1))!}$$ where in the second equality we multiplied and divided by $N-n$, and we used the fact that: $(n+1)!=(n+1)n!$
But: $$(N-n)(N-(n+1))!=(N-n)((N-n)-1)!=(N-n)!$$
So in the end: $${N\choose n+1}=\frac{N-n}{n+1}\frac{N!}{n!(N-n)(N-(n+1))!}=\frac{N-n}{n+1}\frac{N!}{n!(N-n)!}=\frac{N-n}{n+1}{N\choose n}$$