組み合わせの計算をPythonで作ってみる ref: http://qiita.com/kyoro1/items/c9c58496ad3dd798c665

file0.txt

_nC_r=\frac{n!}{r!(n-r)!}

file1.txt

_0C_1=0

file10.txt

_nC_r=_{n-1}C_{r-1}+_{n-1}C_r

file11.py

def comb2(n,r):
    if n < 0 or r < 0 or n < r: 
        return 0
    elif n == 0 or r == 0: 
        return 1
    return comb2(n-1,r-1)+comb2(n-1,r)

file12.py

>>> comb1(20,5)
15504.0
>>> comb2(20,5)
15504

file13.py

>>> import scipy.misc as scm
>>> scm.comb(20, 5)
15504.0

file2.txt

_nC_r=_nC_{r-1}\times\frac{n-r+1}{r}

file3.txt

_nC_r=\frac{n!}{r!(n-r)!}=\frac{n!}{r\times(r-1)!\times(n-r)!}=\frac{n!}{(r-1)!\times(n-r)!}\times\frac{1}{r}\\
_nC_{r-1}=\frac{n!}{(r-1)!\times(n-r+1)\times(n-r)!}=\frac{n!}{(r-1)!\times(n-r)!}\times\frac{1}{n-r+1}

file4.txt

r\times_nC_r=(n-r+1)_nC_{r-1}

file5.txt

_nC_r=_{n-1}C_{r-1}+_{n-1}C_r

file6.txt

\begin{eqnarray}
_nC_r+_nC_{r-1} &=& \frac{n!}{r!(n-r)!}+\frac{n!}{(r-1)!(n-r+1)!}\\
&=& \frac{n!\times\{(n-r+1)+r\}}{r!(n-r+1)!}\\
&=& \frac{n!\times(n+1)}{r!(n-r+1)!}=\frac{(n+1)!}{r!(n+1-r)!}\\
&=& _{n+1}C_r
\end{eqnarray}

file7.txt

_nC_r=_nC_{r-1}\times\frac{n-r+1}{r}

file8.txt

_0C_r=_nC_0=1

file9.py

def comb1(n, r):
    if n == 0 or r == 0: return 1
    return comb1(n, r-1) * (n-r+1) / r