#multinomial

HOWEVER.

Hey, let's talk about names!

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Multinomial coefficients and Pascal's Simplex

For binomials [ (a+b)ⁿ ] the use of Pascal's triangle is helpful. In Pascal's triangle each single row in the triangle defines the coefficients of binomials of each n-value.

For trinomials [ (a+b+c)ⁿ ] this pattern can be extended to a 3-dimensional Pascal's tetrehedron, where each level (and hence a complete triangle) in that tetrehedron defines the coefficients of trinomials of each value of n.

But I do use multiple names wherever I can: One name for this blog, one for my personal side blog, and one in person. Maybe someday I'll take on a second name to use in person based on how my gender might feel that day. Who knows!

This chapter studies a prominent class of recursively-defined combinatorial functions, namely, the binomial and multinomial coefficients and probabilities. The chapter reviews the basic notions and mathematical definitions of these four functions. Subsequently, it characterizes each of these functions via a recursive relation that is valid over a certain two-dimensional or multi-dimensional region and is supplemented with certain boundary conditions. Visual interpretations of these characterizations are given in terms of regular acyclic signal flow graphs. The graph for the binomial coefficients resembles a Pascal Triangle, while that for trinomial or multinomial coefficients looks like a Pascal Pyramid, Tetrahedron, or Hyper-Pyramid. Each of the four functions is computed using both its conventional and recursive definitions. Moreover, the recursive structures of the binomial coefficient and the corresponding probability are utilized in an iterative scheme, which is substantially more efficient than the conventional or recursive evaluation. Analogous iterative evaluations of the multinomial coefficient and probability can be constructed similarly. Applications to the reliability evaluation for two-valued and multi-valued k-out-of-n systems are also pointed out.  

Author  Details:

Ali Muhammad Ali Rushdi

Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.

Mohamed Abdul Rahman Al-Amoudi

Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.

Read full article: http://bp.bookpi.org/index.php/bpi/catalog/view/46/226/386-1

original source : http://onlinestatbook.com/2/probability/multinomial.html