I'm reading Lang's Undergraduate Analysis:
Let ${n}\choose{k}$ denote the binomial coefficient,
$${n\choose k}=\frac{n!}{k!(n-k)!}$$
where $n,k$ are integers $\geq0,0\leq k\leq n$, and $0!$ is defined to be $1$. Prove the following assertion:
$${n\choose k}={n\choose n-k}$$
I proceded by making the adequate substitutions:
$${n\choose n-k}=\frac{n!}{\color{red}{(n-k)}!(n-\color{red}{(n-k)})!}$$
And then I simplified and achieved:
$$\frac{n!}{(n-k)!-k!}$$
But I'm not sure on how to proceed from here, I've noticed that this result is very similar to $\frac{n!}{k!(n-k)!}$. What should I do? I guess it has something to do with the statement about the nature of $n$ and $k$:
$n,k$ are integers $\geq0,0\leq k\leq n$
So should I just change the minus sign to plus sign and think of it as a product of $(n-k)!$?
$$\frac{n!}{(n-k)!-k!}\Rightarrow\frac{n!}{(n-k)!+k!}\Rightarrow \frac{n!}{k!(n-k)!}$$
I'm in doubt because I've obtained the third result on Mathematica, but obtained the first with paper and pencil. I'm not sure if there are different rules for simplification with factorials. I'm not sure if this $(n-k)!+k!$ mean a sum or a product in this case.
Best Answer
You simply failed to distribute the negative sign (i.e., multiplying the difference by $-1$: $$\begin{align} n - (n - k) &= n + -1(n -k) \\ \\& = n + - 1\cdot n - (-1)\cdot k \\ \\ & = n - n + k \\ \\ &= k\end{align}$$
(I.e., Negation distributes over sums and differences.)
$${n\choose n-k}=\frac{n!}{\color{red}{(n-k)}!(n-\color{red}{(n-k)})!} =\frac{n!}{(n-k)!(n - n + k))!} =\frac{n!}{(n-k)!(k)!}= \frac{n!}{k!(n-k)!}$$