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If there are k positions in a sequence, and n1 distinguishable elements can occupy position 1, n2 can occupy position 2, ... and nk can occupy position k, the number of distinct sequences of k elements each is given by
N (k | n1, n2, …, nk) = n1n2nk (B.1)

For example, if a design involves three parameter φ1, φ2, φ3, and there are, respectively, 2, 3, 4 values of these parameters, the number of feasible design is
N (3 | 2, 3, 4) = (2) (3) (4) = 24

In a set of n distinct elements, the number of k-element ordered sequences, or arrangements, is:
(B.2)


For example, if no digits are repeated, the number of four-digit figures is (10)4 = (10) (9) (8) (7) = 5040, whereas, if the digits can be repeated the number of four-digit figures would be (10)4 = 10,000—the latter case would include 0000 as one of the figures.

 
Combination
 
In a set of n distinguishable elements, the number of possible subsets of k different elements each (regardless of order) is given by the binomial coefficient
(B.3)

Equation B.3 is defined only for kn. Using Eq. B.2 for (n)k, we have
(B.3a)

For example, the number of possible samples of four cards each in a deck is:


Excerpt from: Ang, Alfredo H-S. and Tang, Wilson H. Probability Concepts in Engineering Planning and Design. New York: John Wiley & Sons, Inc., 1975.

Copyright © 1975, by John Wiley & Sons, Inc.

This material is used by permission of John Wiley & Sons, Inc.