• $$\large \binom{n}{k}=\frac{n(n-1)\dots(n-k+1)}{k(k-1)\dots(1)} \tag{2}$$

• $$\large \binom{n}{k}=\frac{n!}{k!(n-k)!} \tag{5}$$

• $$\large \binom{n}{k}=\binom{n}{n-k},\;整数n\geq0,整数k \tag{6}$$

• $$\large \binom{r}{k}=\frac{r}{k} \binom{r-1}{k-1},\;整数k\neq 0 \tag{7}$$

• $$\large \binom{r}{k}=\frac{r}{r-k} \binom{r-1}{k},\;整数k\neq r \tag{8}$$

• $$\large \binom{r}{k}=\binom{r-1}{k} + \binom{r-1}{k-1},\;整数k\tag{9}$$

• $$\large \sum_{k=0}^n\binom{r+k}{k}=\binom{r+n+1}{n},\;整数n \geq 0 \tag{10}$$

• $$\large \sum_{k=0}^n\binom{k}{m}=\binom{n+1}{m+1},\;整数m \geq 0,\;整数n \geq 0 \tag{11}$$

• $$\large (x+y)^r=\sum_k\binom{r}{k}k^ry^{r-k},\;整数r \geq 0\tag{13}$$

• $$\large \binom{r}{k}=(-1)^k\binom{k-r-1}{k},\;整数k \tag{17}$$

• $$\large \sum_{k \leq n}\binom{r}{k}=(-1)^n\binom{r-1}{n},\;整数n \tag{18}$$

• $$\large \binom{n}{m}=(-1)^{n-m}\binom{-(m+1)}{n-m},\;整数n\geq 0,\;整数m \tag{19}$$

• $$\large \binom{r}{m}\binom{m}{k}=\binom{r}{k}\binom{r-k}{m-k},\;整数m,\;整数k \tag{20}$$

• $$\large \sum_k\binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n},\;整数n \tag{21}$$

• $$\large \sum_k\binom{r}{m+k}\binom{s}{n+k}=\binom{r+s}{r-m+n},\;整数m,\;整数n,\;整数r\geq 0 \tag{22}$$

• $$\large \sum_k\binom{r}{k}\binom{s+k}{n}(-1)^{r-k}=\binom{s}{n-r},\;整数n,\;整数r\geq 0 \tag{23}$$

• $$\large \sum_{k=0}^r\binom{r-k}{m}\binom{s}{k-t}(-1)^{k-t}=\binom{r-t-s}{r-t-m},\;整数t\geq0,\;整数r\geq 0 \,\;整数m\geq 0 \tag{24}$$

• $$\large \sum_{k=0}^r\binom{r-k}{m}\binom{s+k}{n}=\binom{r+s+1}{m+n+1},\;整数n\geq0,\;整数s\geq 0 \,\;整数m\geq 0,\;整数r\geq 0 \tag{25}$$

• $$\large \sum_{k\geq 0}\binom{r-tk}{k}\binom{s-t(n-k)}{n-k}\frac{r}{r-tk}=\binom{r+s-tn}{n},\;整数n \tag{26}$$