1. 1.

   $n\choose r$ is an integer (proof.).

2. 2.

   ${n\choose r}={n\choose n-r}$.

3. 3.

   ${n\choose r-1}+{n\choose r}={n+1\choose r}$ (Pascal’s rule).

4. 4.

   ${n\choose 0}=1={n\choose n}$ for all $n$.

5. 5.

   ${n\choose 0}+{n\choose 1}+{n\choose 2}+\cdots+{n\choose n}=2^{n}$.

6. 6.

   ${n\choose 0}-{n\choose 1}+{n\choose 2}-\cdots+(-1)^{n}{n\choose n}=0$ for $n>0$.

7. 7.

   $\sum_{{t=k}}^{n}{t\choose k}={n+1\choose k+1}$.

Properties 5 and 6 are the binomial theorem applied to $(1+1)^{n}$ and $(1-1)^{n}$, respectively, although they also have purely combinatorial meaning.

Suppose $n\geq r$ are integers. The below list shows some examples where the binomial coefficients appear.

• $n\choose r$ constitute the coefficients when expanding the binomial $(x+y)^{n}$ – hence the name binomial coefficients. See Binomial Theorem.

• $n\choose r$ is the number of ways to choose $r$ objects from a set with $n$ elements.

• On the context of computer science, it also helps to see ${n\choose r}$ as the number of strings consisting of ones and zeros with $r$ ones and $n-r$ zeros. This equivalency comes from the fact that if $S$ be a finite set with $n$ elements, ${n\choose r}$ is the number of distinct subsets of $S$ with $r$ elements. For each subset $T$ of $S$, consider the function

  $$X_{T}\colon S\rightarrow\{0,1\}$$

  where $X_{T}(x)=1$ whenever $x\in T$ and $0$ otherwise (so $X_{T}$ is the characteristic function for $T$). For each $T\in\mathcal{P}(S)$, $X_{T}$ can be used to produce a unique bit string of length $n$ with exactly $r$ ones.

The ${n\choose r}$ notation was first introduced by von Ettinghausen [1] in 1826, altough these numbers have been used long before that. See this page for some notes on their history. Although the standard mathematical notation for the binomial coefficients is $n\choose r$, there are also several variants. Especially in high school environments one encounters also ${C}(n,r)$ or ${C}^{n}_{r}$ for ${n\choose r}$.

Remark. It is sometimes convenient to set ${n\choose r}:=0$ when $r>n$. For example, property 7 above can be restated: $\sum_{{t=1}}^{n}{t\choose k}={n+1\choose k+1}$. It can be shown that ${n\choose r}$ is elementary recursive.