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

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

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

Multinomial Theorem

For any positive integer $i$ and any nonnegative integer $n$ $(x_{1} + \dots + x_{i})^{n} = \sum_{k_{1} + \dots + k_{i} = n} (k _{1} , \dots , k _{i} n) x_{1}^{k_{1}} \dots x_{i}^{k_{i}}$

In the case $i = 2$ , the multinomial theorem reduces to the binomial theorem.

Binomial Theorem

$(a + b)^{n} = \sum_{k = 0}^{n} (k n) a^{n - k} b^{k} = \sum_{k = 0}^{n} (k n) a^{k} b^{n - k}$ Special case: $(1 + x)^{n} = \sum_{k = 0}^{n} (k n) x^{k}$

Pascal’s Rule

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

Explorer

Binomial Coefficient

Multinomial Coefficient

Multinomial Theorem

Binomial Theorem

Pascal’s Rule

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