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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.

RobPratt
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Sam Shepherd
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    Use the definition of binomial coefficients in terms of factorials: $\binom{N}{n+1} = \frac{N!}{(n+1)!(N-n-1)!} = \frac{N(N-1)(N-2) \cdots (N-n)}{(n+1)!}$, and then factor out the appropriate factors in the numerator and denominator. – Nick May 06 '23 at 01:52

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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: $${N\choose n+1}=\frac{N-n}{n+1}\frac{N!}{n!(N-n)(N-(n+1))!}=\frac{N-n}{n+1}\frac{N!}{n!(N-n)!}=\frac{N-n}{n+1}{N\choose n}$$

Fotis
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