User:Racvets1/Sandbox2

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{k \choose n}=\frac{n!}{k!(n-k)!}

(a+b)^{n}={n \choose 0}a^{n}b^{0}+{n \choose 1}a^{n-1}b^{1}+{n \choose 2}a^{n-2}b^{2}+\cdots+{n \choose n-1}a^{1}b^{n-1}+{n \choose n}a^{0}b^{n}

(a+b)^{n}=\sum_{j=0}^{n}{n \choose j}a^{n-j}b^{j}

S_{k}=a^0+a^1+\cdots+a^k=\sum_{r=0}^{k}a^{r}=\frac{a^{k+1}-1}{a-1}

k\rightarrow\infty,\left|a\right|<1

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

{n \choose 0}-{n \choose 1}+{n \choose 2}-\cdots+(-1)^n{n \choose n}=0

1+2+3+\cdots+n=\frac{n(n+1)}{2}