Interesting Binomial Coefficients

\[\binom{n}{k} = \frac{n(n-1)\ldots(n-k+1)}{k(k-1)\ldots(1)}\]
\[\]

Binomial coefficients have a lot of interesting properties. The following are some of them.

Most of the formulas are important. Some of them are very important. They deserve our careful studies.

  1. \(\binom{n}{k} = \binom{n}{n-k}\)

    a.jpg
    \[\begin{align} \binom{n}{k} &= \binom{n}{n-k},\qquad {\rm integer}\; n\ge 0,\; {\rm integer}\; k. \\ \binom{5}{1} &= \binom{5}{5-1} \end{align}\]
    \[\]

  2. \(\binom{r}{k}=\frac{r}{k}\binom{r-1}{k-1}\)

    b.jpg
    \[\begin{align} \binom{r}{k} &= \frac{r}{k}\binom{r-1}{k-1},\qquad {\rm integer}\; k\neq 0. \\ \binom{7}{2} &= \frac{7}{2}\binom{7-1}{2-1} \end{align}\]
    \[\]

  3. \(\binom{r}{k} = \frac{r}{r-k}\binom{r-1}{k}\)

    c.jpg
    \[\begin{align} \binom{r}{k} &= \frac{r}{r-k}\binom{r-1}{k},\qquad {\rm integer}\; k\neq r. \\ \binom{7}{3} &= \frac{7}{7-3}\binom{7-1}{3} \end{align}\]
    \[\]

  4. \(\binom{r}{k} = \binom{r-1}{k} + \binom{r-1}{k-1}\)

    d.jpg
    \[\begin{align} \binom{r}{k} &= \binom{r-1}{k} + \binom{r-1}{k-1},\qquad {\rm integer}\; k. \\ \binom{7}{3} &= \binom{6}{3} + \binom{6}{2} \end{align}\]
    \[\]

  5. \(\sum_{0\leq k\leq n}\binom{r+k}{k}=\binom{r}{0}+\binom{r+1}{1}+\cdots+\binom{r+n}{n}=\binom{r+n+1}{n}\)

    e.jpg
    \[\begin{align} \sum_{0\leq k\leq n}\binom{r+k}{k} &= \binom{r}{0}+\binom{r+1}{1}+\cdots+\binom{r+n}{n}=\binom{r+n+1}{n},\qquad{\rm integer}\; n\geq0. \\ \sum_{0\leq k\leq 3}\binom{4+k}{k} &= \binom{4}{0}+\binom{4+1}{1}+\cdots+\binom{4+3}{3}=\binom{4+3+1}{3} \end{align}\]
    \[\]

  6. \(\sum_{0\leq k\leq n}\binom{k}{m}=\binom{0}{m}+\binom{1}{m}+\cdots+\binom{n}{m}=\binom{n+1}{m+1}\)

    f.jpg
    \[\begin{align} \sum_{0\leq k\leq n}\binom{k}{m} &= \binom{0}{m}+\binom{1}{m}+\cdots+\binom{n}{m}=\binom{n+1}{m+1},\qquad{\rm integer}\; m\geq0,\;{\rm integer}\; n\geq0. \\ \sum_{0\leq k\leq 7}\binom{k}{2} &= \binom{0}{2}+\binom{1}{2}+\cdots+\binom{7}{2}=\binom{7+1}{2+1} \end{align}\]
    \[\]

  7. \(\binom{-r}{k}=(-1)^{k}\binom{r+k-1}{k}\)

    g.jpg
    \[\begin{align} \binom{-r}{k} &= (-1)^{k}\binom{r+k-1}{k},\qquad{\rm integer}\; k. \\ \binom{-3}{5} &= (-1)^{5}\binom{3+5-1}{5} \end{align}\]
    \[\]

  8. \(\sum_{k\leq n}\binom{r}{k}(-1)^{k}=\binom{r}{0}-\binom{r}{1}+\cdots+(-1)^{n}\binom{r}{n}=(-1)^{n}\binom{r-1}{n}\)

    h.jpg
    \[\begin{align} \sum_{k\leq n}\binom{r}{k}(-1)^{k} &= \binom{r}{0}-\binom{r}{1}+\cdots+(-1)^{n}\binom{r}{n}=(-1)^{n}\binom{r-1}{n},\qquad{\rm integer}\; n\geq0. \\ \sum_{k\leq 3}\binom{7}{k}(-1)^{k} &= \binom{7}{0}-\binom{7}{1}+\cdots+(-1)^{3}\binom{7}{3}=(-1)^{3}\binom{7-1}{3} \end{align}\]
    \[\]

  9. \(\binom{n}{m}=(-1)^{n-m}\binom{-(m+1)}{n-m}\)

    i.jpg
    \[\begin{align} \binom{n}{m} &= (-1)^{n-m}\binom{-(m+1)}{n-m},\qquad{\rm integer}\; n\geq0,\;{\rm integer}\; m. \\ \binom{n}{3} &= (-1)^{n-3}\binom{-(3+1)}{n-3} \end{align}\]
    \[\]

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