
If there are k positions in a sequence, and n_{1} distinguishable elements can occupy position 1, n_{2} can occupy position 2, ... and n_{k} can occupy position k, the number of distinct sequences of k elements each is given by

N (k  n_{1}, n_{2}, …, n_{k}) = n_{1}n_{2}…n_{k}_{ }(B.1)

For example, if a design involves three parameter φ_{1}, φ_{2}, φ_{3}, 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 kelement ordered sequences, or arrangements, is:

(B.2)

For example, if no digits are repeated, the number of fourdigit figures is (10)_{4} = (10) (9) (8) (7) = 5040, whereas, if the digits can be repeated the number of fourdigit 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 k ≤ n. 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 HS. 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. 