- ( n − 1 ) × ⋯ × ( n − k + 1 ) k × ( k − 1 ) × ⋯ × 1 , {\displaystyle {\
binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times...
- n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 , {\displaystyle {\
binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},}
which can be written...
- s_{4B}(k)={\
binom {2k}{k}}\sum _{j=0}^{k}4^{k-2j}{\
binom {k}{2j}}{\
binom {2j}{j}}^{2}={\
binom {2k}{k}}\sum _{j=0}^{k}{\
binom {k}{j}}{\
binom {2k-2j}{k-j}}{\
binom {2j}{j}}=1...
- p^{-r}=(1-q)^{-r}=\sum _{k=0}^{\infty }{\
binom {-r}{{\phantom {-}}k}}(-q)^{k}=\sum _{k=0}^{\infty }{\
binom {k+r-1}{k}}q^{k}}
hence the
terms of the probability...
- f(k,n,p)=\Pr(X=k)={\
binom {n}{k}}p^{k}(1-p)^{n-k}} for k = 0, 1, 2, ..., n,
where ( n k ) = n ! k ! ( n − k ) ! {\displaystyle {\
binom {n}{k}}={\frac {n...
- n + 1 r + 1 ) . {\displaystyle {\
binom {r}{r}}+{\
binom {r+1}{r}}+{\
binom {r+2}{r}}+\cdots +{\
binom {n}{r}}={\
binom {n+1}{r+1}}.} The name
stems from...
- ( n k ) + ( n k − 1 ) = ( n + 1 k ) , {\displaystyle {\
binom {n}{k}}+{\
binom {n}{k-1}}={\
binom {n+1}{k}},} by Pascal's identity. On the
other hand, if...
- {\displaystyle {\
binom {b_{1}}{b_{2}}}=x_{1}{\
binom {a_{11}}{a_{21}}}+x_{2}{\
binom {a_{12}}{a_{22}}}} and ( a 12 a 22 ) . {\displaystyle {\
binom {a_{12}}{a_{22}}}...
- {\displaystyle {\frac {{\
binom {a+c}{a}}{\
binom {b+d}{b}}(a+b)!(c+d)!}{n!}}={\frac {{\
binom {a+c}{a}}{\
binom {b+d}{b}}}{\
binom {n}{a+b}}}}
Another derivation:...
- The
Gaussian binomial coefficient,
written as ( n k ) q {\displaystyle {\
binom {n}{k}}_{q}} or [ n k ] q {\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}}...