Show that $\sum_{m=0}^n(-1)^m\binom nm=0$

Question

Show that $$ \sum_{m=0}^n (-1)^m \binom{n}{m} = 0$$

I have first difficulty understanding the summation notation: For example what $ \sum_{m=0}^3 (-1)^m \binom{n}{m}$ would mean? I suppose that it means: $$ \sum_{m=0}^2 (-1)^m \binom{n}{m} = (-1)^0 \binom{2}{0} + (-1)^1 \binom{2}{1} + (-1)^2 \binom{2}{2} $$

$$ \sum_{m=0}^2 (-1)^m \binom{n}{m} = + 1 - 2 + 1 =0 $$

If it is such case, it shows that the value of $(-1)^m$ alternate positive when $m$ is even and negative when $m$ is odd.

Therefore the sum when $m$ is even $+$ the sum when $m$ is odd $= 0$ and we can then factor out the constant $+1$ from the first sum and $-1$ from the second sum and then QED.

Is this approach correct? How to write this using sum notation when $m$ even and when $m$ is odd?
Is there a better approach?

Much appreciated.

Answer 1

the expansion of $(1+(-1))^n=\sum _0 ^n (-1)^m .(1)^{n-m} {n \choose m}$

Answer 2

This is Newton's Binomial Series:

$$\sum_{m=0}^n\binom{n}{m}(-1)^m = \sum_{m=0}^n\binom{n}{m}(-1)^m1^{n-m} = (-1 + 1)^n = 0^n = 0$$

Answer 3

We have, $$(a+b)^n=\sum\limits_{m=0}^n {n \choose m}a^{n-m}b^m$$

Now, put $a=1, b=-1$

$\sum\limits_{m=0}^n {n \choose m}(-1)^m=(1-1)^n=0$

Answer 4

There is a combinatorial proof of this fact too. Suppose $n\geq 1$ (otherwise it is trivial). Let $N$ be a set of $n$ elements. Let $S_0=\{A\subseteq N\colon |A|\,\text{even}\}$ and $S_1=\{A\subseteq N\colon |A|\,\text{odd}\}$. The identity in question is equivalent to $|S_0|=|S_1|$. Fix an element $a\in N$. Define the map $\varphi\colon S_0\to S_1$ by $$ \varphi(A)= \begin{cases} A\cup\{a\}&\text{if $a\notin A$}\\ A\setminus\{a\}&\text{if $a\in A$}. \end{cases} $$ It is easy to see that the map $\varphi$ is a bijection (easy to write down an inverse) and hence $|S_0|=|S_1|$.