AOCP/Multinomial Coefficients - Revision history

Admin: /* Definition */

2017-08-06T10:20:09Z

Definition

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	</math>		</math>

	~~This can also generalize the~~ binomial theorem to ~~a higher power~~ of multiple ~~sums~~:		===Generalization of Binomial Theorem===

			The binomial theorem gives a formula that allows powers of binomial sums to be expanded in terms of binomial coefficients:

			<math>
			(x+y)^n = \sum_{0 \leq k \leq n} \binom{n}{k} x^k y^{(n-k)}
			</math>

			There is an analogous expansion for powers of multinomial sums (sums of multiple terms), in terms of these multinomial coefficients:

	<math>		<math>

Admin: /* Definition */

2017-08-06T10:17:14Z

Definition

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	===Definition===		===Definition===

	We can generalize this approach and define the multinomial coefficient:		We can generalize this approach and define the multinomial coefficient.

			More compactly, let n denote the sum of the ks:

			<math>
			\binom{n}{k_1 , k_2 , \dots , k_m } = \dfrac{n!}{k_1 ! k_2 ! \dots k_m !} \qquad k_i \in \mathbb{Z}, k_i \geq 0
			</math>

			or,

	<math>		<math>

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2017-07-24T08:13:53Z

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	{{AOCPFlag}}		{{AOCPFlag}}

			[[Category:Binomial]]
			[[Category:Multinomial]]
			[[Category:Combinatorics]]
			[[Category:Permutations]]

Admin: /* Related Pages */

2017-07-24T08:13:34Z

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	=Related Pages=		=Related Pages=

	~~See also:~~		[[LatticePaths]]
	* [[LatticePaths]]
	* [[AOCP/Multisets]]		[[AOCP/Multisets]]

			[[AOCP/Binomial Coefficients]]

	=Flags=		=Flags=

	{{AOCPFlag}}		{{AOCPFlag}}

Admin: /* Related Pages */

2017-07-20T06:53:28Z

← Older revision		Revision as of 06:53, 20 July 2017
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	See also:		See also:
	* [[~~LaticePaths~~]]		* [[LatticePaths]]
	* [[AOCP/Multisets]]		* [[AOCP/Multisets]]


	=Flags=		=Flags=

	{{AOCPFlag}}		{{AOCPFlag}}

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2017-07-20T06:53:19Z

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	\binom{k_1 + k_2 + \dots + k_m}{k_1, k_2, \dots k_m} = \binom{k_1 + k_2}{k_1} \binom{k_1 + k_2 + k_3}{k_1 + k_2} \dots \binom{k_1 + k_2 + \dots + k_m}{k_1 + \dots + k_{m-1}}		\binom{k_1 + k_2 + \dots + k_m}{k_1, k_2, \dots k_m} = \binom{k_1 + k_2}{k_1} \binom{k_1 + k_2 + k_3}{k_1 + k_2} \dots \binom{k_1 + k_2 + \dots + k_m}{k_1 + \dots + k_{m-1}}
	</math>		</math>

			=Related Pages=

			See also:
			* [[LaticePaths]]
			* [[AOCP/Multisets]]


	=Flags=		=Flags=

	{{AOCPFlag}}		{{AOCPFlag}}

Admin: Created page with "=Volume 1= ==Chapter 1: Basic Concepts: Multinomial Coefficients== ===Definition=== We can generalize this approach and define the multinomial coefficient: $\binom{k..."$

2017-07-20T06:52:51Z

Created page with "=Volume 1= ==Chapter 1: Basic Concepts: Multinomial Coefficients== ===Definition=== We can generalize this approach and define the multinomial coefficient: <math> \binom{k..."

New page

=Volume 1=

==Chapter 1: Basic Concepts: Multinomial Coefficients==

===Definition===

We can generalize this approach and define the multinomial coefficient:

<math>
\binom{k_1 + k_2 + \dots + k_m}{k_1 , k_2 , \dots , k_m } = \dfrac{(k_1 + k_2 + \dots + k_m)!}{k_1 ! k_2 ! \dots k_m !} \qquad k_i \in \mathbb{Z}, k_i \geq 0
</math>

This can also generalize the binomial theorem to a higher power of multiple sums:

<math>
(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n} \binom{n}{k_1, k_2, \dots, k_m} x_1^{k_1} x_2^{k_2} \dots x_m^{k_m}
</math>

Any multinomial coefficient can also be expressed in terms of binomial coefficients:

<math>
\binom{k_1 + k_2 + \dots + k_m}{k_1, k_2, \dots k_m} = \binom{k_1 + k_2}{k_1} \binom{k_1 + k_2 + k_3}{k_1 + k_2} \dots \binom{k_1 + k_2 + \dots + k_m}{k_1 + \dots + k_{m-1}}
</math>

=Flags=

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AOCP/Multinomial Coefficients - Revision history

Admin: /* Definition */

Admin: /* Definition */

Admin: /* Flags */

Admin: /* Related Pages */

Admin: /* Related Pages */

Admin: /* Flags */

Admin: Created page with "=Volume 1= ==Chapter 1: Basic Concepts: Multinomial Coefficients== ===Definition=== We can generalize this approach and define the multinomial coefficient: \binom{k..."

Admin: Created page with "=Volume 1= ==Chapter 1: Basic Concepts: Multinomial Coefficients== ===Definition=== We can generalize this approach and define the multinomial coefficient: $\binom{k..."$