Handbook of Civil Engineering Calculations

this class, and then assign integers to the remaining classes in consecutive and ascending

order, as shown in column 4 of Table 23. These integers are the class codes. Let c = class

code, w = class width, and A = midpoint of class having the code O. Compute X/c, or 2/c

= 3(-2) + 9(-l) + 7(0) + 5(1) - -10. Now set X = A + w@fc)/n> or X =^30 + 4(-10)/24 =

28.33 min.

5. Compute the standard deviation by the coding method
   Using the codes previously assigned, compute 2/c
   2
   , or 2/c
   2
   = 3(-2)
   2
   
   
   • 9(-l)
     2
     
     
     • 7(O)
       2
       
       
       • 5(1)
         2
         = 26. Now set s
         2
         = w*{(2fi?yn - [&fc)/n]
         2
         }. Then s
         2
         = 16[26/24 - (-10/24)
         2
         ] =
         14.5556, and s = V14.5556 = 3.82 min.
       
       
       
       
     
     
     
     
   
   
   
   

Permutations and Combinations

An arrangement of objects or individuals in which the order or rank is significant is called
a permutation. A grouping of objects or individuals in which the order or rank is not sig-
nificant, or in which it is predetermined, is called a combination. Assume that n objects
are available and that r of these objects are selected to form a permutation or combination.
If interest centers on only the identity of the r objects selected, a combination is formed; if
interest centers on both the identity and the order or rank of the r objects, a permutation is
formed. In the following material, the r objects all differ from one another.
Where necessary, the number of permutations or combinations that can be formed is
computed by applying the following law, known as the multiplication law: If one task can
be performed in ml different ways and another task can be performed in m 2 different
ways, the set of tasks can be performed in W 1 W 2 different ways.

Notational System

Here n\ (read "w factorial" or "factorial n") = product of first n integers, and the integers

are usually written in reverse order. Thus, 5! = 5x4*3x2x 1 = 120. For mathematical

consistency, O! is taken as 1.

Also, Pn^ = number of permutations that can be formed of n objects taken r at a time;

and Cn r = number of combinations that can be formed of n objects taken r at a time.

NUMBER OF WAYS OFASSIGNING WORK

A firm has three machines, A, B, and C, and each machine can be operated by only one

individual at a time. The number of employees who are qualified to operate a machine is:

machine A, five; machine B, three; machine C, seven. In addition to these 15 employees,

Smith is qualified to operate all three machines. In how many ways can operators be as-

signed to the machines?

Calculation Procedure:

1. Compute the number of possible assignments if Smith
   is excluded
   Apply the multiplication law. The number of possible assignments = 5x3x7=105.