Applied Statistics and Probability for Engineers

2-2

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

ing the element to be placed in the first position of the sequence from the nelements, then

selecting the element for the second position from the n 1 remaining elements, then select-

ing 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.

The number of permutations of a subset of relements selected from a set of ndiffer-

ent elements is

Prnn 1 n 12  1 n 22 p 1 nr 12  (S2-2)

n!

1 nr 2!

The number of permutations of objects of which n 1 are of

one type, n 2 are of a second type, , and nrare of an rth type is

(S2-3)

n!

n 1! n 2! n 3! p nr!

p

nn 1 n 2 p nr

EXAMPLE S2-2 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 first compo-

nent, a location from the remaining seven for the second component, a location from the re-

maining six for the third component, and a location from the remaining five for the fourth

component. Therefore,

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.

P 48  8  7  6  5 

8!

4!

1680 different designs are possible.

EXAMPLE S2-3 Consider a machining operation in which a piece of sheet metal needs two identical diameter

holes drilled and two identical size notches cut. We denote a drilling operation as dand a

notching operation as n. In determining a schedule for a machine shop, we might be interested

in the number of different possible sequences of the four operations. The number of possible

sequences for two drilling operations and two notching operations is

The six sequences are easily summarized: ddnn,dndn,dnnd,nddn,ndnd,nndd.

4!

2! 2!

 6

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