( 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...

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...

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...

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}}...

( 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...

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...

{\binom {m-1}{k-1}}{1}}\sum _{n=m}^{\infty }{1 \over {\binom {n}{k}}}\\&={\frac {\binom {m-1}{k-1}}{1}}\cdot {\frac {k}{k-1}}\cdot {\frac {1}{\binom...

{\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:...

) ( N n ) , {\displaystyle p_{X}(k)=\Pr(X=k)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}},} where N {\displaystyle N} is the population...

= ( 10 − 1 4 − 1 ) = ( 9 3 ) = 84. {\displaystyle {\binom {n-1}{k-1}}={\binom {10-1}{4-1}}={\binom {9}{3}}=84.} This corresponds to compositions of an...

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...

binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset...

{\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}}}...

_{k=0}^{n}{\binom {n}{k}}f^{(n+1-k)}g^{(k)}+\sum _{k=1}^{n+1}{\binom {n}{k-1}}f^{(n+1-k)}g^{(k)}\\&={\binom {n}{0}}f^{(n+1)}g^{(0)}+\sum _{k=1}^{n}{\binom...

0=(1-1)^{t}={\binom {t}{0}}-{\binom {t}{1}}+{\binom {t}{2}}-\cdots +(-1)^{t}{\binom {t}{t}}.} Using the fact that ( t 0 ) = 1 {\displaystyle {\binom {t}{0}}=1}...

}{\binom {1}{m}}{\binom {1}{n-m}}\\&=p^{n}\left(1-p\right)^{2-n}\left[{\binom {1}{0}}{\binom {1}{n}}+{\binom {1}{1}}{\binom {1}{n-1}}\right]\\&={\binom...

{\begin{aligned}{\binom {n-k}{k}}+{\binom {n-k-1}{k-1}}&={\frac {n}{n-k}}{\binom {n-k}{k}}\\{\binom {n}{k}}-{\binom {n}{k-1}}&={\frac {n+1-k}{n+1-2k}}{\binom {n}{k}}\\{\binom...

value: Pr ( X = k ) = ( n k ) p k ( 1 − p ) n − k {\displaystyle \Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}} If the null hypothesis H 0 {\displaystyle H_{0}}...

}\left(-1\right)^{k}{\binom {n}{k}}{\binom {2n-2k}{n}}x^{n-2k},\\[1ex]P_{n}(x)&=2^{n}\sum _{k=0}^{n}x^{k}{\binom {n}{k}}{\binom {\frac {n+k-1}{2}}{n}}...

}}\right]\\&=(n-1)!{\frac {n}{k!(n-k)!}}\\&={\frac {n!}{k!(n-k)!}}\\&={\binom {n}{k}}.\end{aligned}}} Pascal's rule can be generalized to multinomial...

n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}-{\binom {m+1}{1}}B_{1}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}-\cdots +(-1)^{m}{\binom {m+1}{m}}B_{m}n\right)} for all sums...

) = ( n − 1 k − 1 ) + ( n − 1 k ) , {\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}},} with the base cases ( n 0 ) = ( n n ) = 1...

) ! {\displaystyle {\binom {\alpha }{\beta }}={\binom {\alpha _{1}}{\beta _{1}}}{\binom {\alpha _{2}}{\beta _{2}}}\cdots {\binom {\alpha _{n}}{\beta _{n}}}={\frac...

2 ) . {\displaystyle {\binom {n}{\tfrac {n+m}{2}}}-{\binom {n}{{\tfrac {n+m}{2}}+1}}={\frac {m+1}{{\tfrac {n+m}{2}}+1}}{\binom {n}{\tfrac {n+m}{2}}}.}...

( d + 1 ) n − 1 ¯ . {\displaystyle \left(\!\!{\binom {n}{d}}\!\!\right)={\binom {n+d-1}{d}}={\binom {d+(n-1)}{n-1}}={\frac {(d+1)\times (d+2)\times \cdots...

( 2 n n ) 2 {\displaystyle \sum _{k=0}^{n}{\binom {2k}{k}}{\binom {2n-2k}{n-k}}{\binom {2n}{2k}}={\binom {2n}{n}}^{2}} (sequence A002894 in the OEIS)...

the basis ( x 0 ) , ( x 1 ) , ( x 2 ) , … {\textstyle {\binom {x}{0}},{\binom {x}{1}},{\binom {x}{2}},\dots } are thus described by similar formulas:...

_{N}(x)&=x^{N+1}\sum _{n=0}^{N}{\binom {N+n}{n}}{\binom {2N+1}{N-n}}(-x)^{n}\qquad N\in \mathbb {N} \\&=\sum _{n=0}^{N}(-1)^{n}{\binom {N+n}{n}}{\binom {2N+1}{N-n}}x^{N+n+1}\\&=\sum...

≡ ∏ i = 0 k ( m i n i ) ( mod p ) , {\displaystyle {\binom {m}{n}}\equiv \prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}},} where m = m k p k + m k...