Recursive Computation of Binomial and Multinomial Coefficients and Probabilities

Recursive Computation of Binomial and Multinomial Coefficients and Probabilities | Chapter 07 | Advances in Mathematics and Computer Science Vol. 1

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.

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