Page 78 - Applied Statistics And Probability For Engineers

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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

9!
2520
4! 3! 2!

Combinations
Another counting problem of interest is the number of subsets of r elements that can be se-
lected from a set of n elements. Here, order is not important. Every subset of r elements 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 r blanks 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
* *

The number of subsets of size r that can be selected from a set of n elements is
n
denoted as 1 r 2 or C r n and
n n!
a b (S2-4)
r r!1n r2!

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

8!
56
5! 3!

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

3 3!
a b 3 different ways
2 2! 1!