Applied Statistics and Probability for Engineers

2-3

EXAMPLE S2-4 A part is labeled by printing with four thick lines, three medium lines, and two thin lines. If

each ordering of the nine lines represents a different label, how many different labels can be

generated by using this scheme?

From Equation S2-3, the number of possible part labels is

Combinations

Another counting problem of interest is the number of subsets of relements that can be se-

lected from a set of nelements. Here, order is not important. Every subset of relements can

be indicated by listing the elements in the set and marking each element with a “*” if it is to

be included in the subset. Therefore, each permutation of r*’s and n rblanks indicate a dif-

ferent subset and the number of these are obtained from Equation S2-3.

For example, if the set is S= {a, b, c, d} the subset {a, c} can be indicated as

abcd

**

9!

4! 3! 2!

 2520

The number of subsets of size rthat can be selected from a set of nelements is

denoted as or and

a (S2-4)

n

r

b

n!

r! 1 nr 2!

1 nr 2 Cnr

EXAMPLE S2-5 A printed circuit board has eight different locations in which a component can be placed. If

five identical components are to be placed on the board, how many different designs are pos-

sible?

Each design is a subset of the eight locations that are to contain the components. From

Equation S2-4, the number of possible designs is

The following example uses the multiplication rule in combination with Equation S2-4 to an-

swer a more difficult, but common, question.

EXAMPLE S2-6 A bin of 50 manufactured parts contains three defective parts and 47 nondefective parts. A

sample of six parts is selected from the 50 parts. Selected parts are not replaced. That is, each

part can only be selected once and the sample is a subset of the 50 parts. How many different

samples are there of size six that contain exactly two defective parts?

A subset containing exactly two defective parts can be formed by first choosing the

two defective parts from the three defective parts. Using Equation S2-4, this step can be

completed in

a

3

2

b

3!

2! 1!

3 different ways

8!

5! 3!

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