Page 46 - Applied statistics and probability for engineers

24 Chapter 2/Probability

This result follows from the multiplication rule. A permutation can be constructed by

selecting the element to be placed in the irst position of the sequence from the n ele-
ments, then selecting the element for the second position from the remaining n −1 ele-
ments, then selecting the element for the third position from the remaining n − 2 elements,
and so forth. Permutations such as these are sometimes referred to as linear permutations.
In some situations, we are interested in the number of arrangements of only some of the
elements of a set. The following result also follows from the multiplication rule.
Permutations
of Subsets The number of permutations of subsets of r elements selected from a set of n
different elements is
n
−
n
P r = ×( n − ) ×( n − ) × ... ×( n r + ) = n! (2-2)
2
1
1
( n r)!
−
Example 2-10 Printed Circuit Board A printed circuit board has eight different locations in which a component can
be placed. If four different components are to be placed on the board, how many different designs are
possible?
Each design consists of selecting a location from the eight locations for the irst component, a location from

the remaining seven for the second component, a location from the remaining six for the third component, and a

location from the remaining ive for the fourth component. Therefore,
P 4 = × × × = 8!
8
8 7 6 5
4!
= 1680 different designsare possible .
Sometimes we are interested in counting the number of ordered sequences for objects that
are not all different. The following result is a useful, general calculation.
Permutations of
Similar Objects
The number of permutations of n = n + n 2 + … + n r objects of which n 1 are of one
1
type, n 2 are of a second type, … , and n r are of an rth type is
n!
(2-3)
n n n ! !... !
!
1
3
n r
2
Example 2-11 Hospital Schedule A hospital operating room needs to schedule three knee surgeries and
two hip surgeries in a day. We denote a knee and hip surgery as k and h, respectively. The
number of possible sequences of three knee and two hip surgeries is
5! = 10
2 3!
!
The 10 sequences are easily summarized:
,
,
,
,
,
,
,
,
{kkkhh kkhkh kkhhk khkkh khkhk khhkk hkkkh hkkhk hkhkk hhkkkk}
,
Example 2-12 Bar Code 39 Code 39 is a common bar code system that consists of narrow and wide bars
(black) separated by either wide or narrow spaces (white). Each character contains nine elements

(ive bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white)
space appears between each bar. The original speciication (since revised) used exactly two wide bars and one wide

space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote
narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). One character is held back as a start
and stop delimiter. How many other characters can be coded by this system? Can you explain the name of the system?