- − k + j j ) . {\displaystyle {\
binom {n}{k}}{\
binom {k}{j}}={\
binom {n}{j}}{\
binom {n-j}{k-j}}={\
binom {n}{k-j}}{\
binom {n-k+j}{j}}.} For
constant n, we...
- 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...
- 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...
- 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...
- ) ( 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 po****tion...
- the
formula is symmetrical, ( n k ) = ( n n − k ) . {\textstyle {\
binom {n}{k}}={\
binom {n}{n-k}}.} A
simple variant of the
binomial formula is obtained...
- ≡ ∏ 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...
-
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...
- cups chosen,
there are ( 8 4 ) = 8 ! 4 ! ( 8 − 4 ) ! = 70 {\displaystyle {\
binom {8}{4}}={\frac {8!}{4!(8-4)!}}=70}
possible combinations. The frequencies...
