Binomial Coefficient

$(k n) = \frac{n !}{k ! ( n - k )!} = \frac{n \times ( n - 1 ) \times \dots \times ( n - k + 1 )}{k \times ( k - 1 ) \times \dots \times 1}$

Symmetry Rule for Binomial Coefficients - $(k n) = (n - k n) for 0 \leq k \leq n$
Zero Choose Zero - $(0 0) = 1$
Binomial Coefficient with One or Self minus One - $(1 n) = (n - 1 n) = n$
Binomial Coefficient with Zero or with Self - $(0 n) = (n n) = 1 for n \geq 0$
Sums of the binomial coefficients - $\sum_{k = 0}^{n} (k n) = 2^{n}$

Multinomial coefficient

$(k _{1} , k _{2} , \dots , k _{i} n) = \frac{n !}{k _{1} ! k _{2} ! \dots k _{i} !} = (k _{1} k _{1}) (k _{2} k _{1} + k _{2}) \dots (k _{i} k _{1} + k _{2} + \dots + k _{i})$

See also: Permutations of multisets

Pascal’s rule

Also called Pascal’s-formula or Pascal’s Recurrence or זהות פסקל $(k n) = (k n - 1) + (k - 1 n - 1)$

Binomial Theorem

$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k} = \sum_{k = 0}^{n} (k n) a^{k} b^{n - k}$ Binomial Theorem for $(1 + x)^{n}$ $(1 + x)^{n} = \sum_{k = 0}^{n} (k n) x^{k}$

Compositions of n

The k-composition of $n$ , (the number ways of writing $n$ as the sum of a sequence in length $k$ of (strictly) positive integers.) is $(k - 1 n - 1)$ .
Total number of compositions^[Also see: https://en.wikipedia.org/wiki/Composition_(combinatorics)#Number_of_compositions] of $n$ : $\sum_{k = 1}^{n} (k - 1 n - 1) = 2^{n - 1}$

Explorer

Binomial Coefficient

Multinomial coefficient

Pascal’s rule

Binomial Theorem

Compositions of n

Table of Contents

Backlinks