# Is this proof of $\binom{n}{k}+\binom{n}{k-1} = \binom{n+1}{k}$ valid?

I want to show that : $$\binom{n}{k}+\binom{n}{k-1} = \frac{(n+1)!}{k!(n+1-k)!}$$

Here is my proof : $$\forall 1\leq k\leq n$$ :

\begin{align} \binom{n}{k}+\binom{n}{k-1} &= \frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n+1-k)!} \tag1 \\[4pt] &=\frac{n!(n+1-k)}{k!(n+1-k)!} +\frac{n!k}{k!(n+1-k)!} \tag2 \\[4pt] &= \frac{n!(n+1-k+k)}{k!(n+1-k)!} \tag3 \\[4pt] &=\frac{(n+1)!}{k!(n+1-k)!} \tag4 \\[4pt] &=\binom{n+1}{k} \tag5 \end{align}

And normally I avoided problem of factorial not defined since $$k\leq n$$.

Do you think this is correct ?

Thank you a lot

EDIT : Thank you everyone and especially Riemann and Jean-Claude Arbault since it is not defined for $$k=0$$...

• I think it's fine, except that you should assume $k\ge 1$, because $\binom{n}{k-1}$ may not be defined if $k=0$. Nov 23, 2022 at 8:15
• this looks good to me Nov 23, 2022 at 8:16
• Your title should probably be ${n\choose k}+{n\choose k-1}={n+1\choose k}$ as that's the more meaningful expression. But your proof, which is just arithmetical manipulation, is just fine. Nov 23, 2022 at 20:40
• It's done ! Thank you ! Nov 23, 2022 at 21:31

Yes it's correct $$\binom{n}k+\binom{n}{k-1}$$ $$=\frac{n!}{(n-k)!\cdot k!}+\frac{n!}{(n+1-k)!\cdot (k-1)!}$$ $$=\frac{n!}{(n-k)!\cdot (k-1)!}\left(\frac{1}{k}+\frac{1}{n+1-k}\right)$$ $$=\frac{n!}{(n-k)!\cdot (k-1)!}\left(\frac{n+1-k+k}{k(n+1-k)}\right)$$ $$=\frac{n!(n+1)}{(n+1-k)!\cdot k!}$$ $$=\frac{(n+1)!}{(n+1-k)!\cdot k!}$$ $$=\binom{n+1}{k}$$ Here we made the assumption that $$k\ne 0$$
• One must be careful with boundaries. (a well-known trap for induction proofs, for instance, though it's not the case here) As stated by Rieman above, there is an error in the proof. For $k=0$, $n\choose k-1$ is not defined. And even if you care to define it to be zero, $(-1)!$ is not defined, and there is no way you are going to define it to be zero. Nov 23, 2022 at 10:12
• It's understood that $k\ne0$ Nov 23, 2022 at 10:16
• Btw isn't $(-1)!=(0-1)!=\Gamma(0)?$ Nov 23, 2022 at 10:18