Greenwood press

COUNTING
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emergency number. Therefore there are a total of 
8 × 10 × 10 − 1 = 799 possi-
ble area codes. The local phone number has seven digits, with a three-digit pre-
fix and four-digit suffix. In the prefix, the first digit cannot be 0 or 1 for the same
reason mentioned earlier. Also, the prefix cannot use 555, because that is a
dummy set of numbers used in entertainment media, such as movies and songs,
except for the national information number, 555-1212. Therefore the prefix can
have
8 × 10 × 10 − 1 = 799 possible values. The suffix can have any four-digit
number, which means there are 
10 × 10 × 10 × 10 = 10,000 possible values.
Therefore, using the multiplication-counting principle, there is a total of 
799 ×
799 × 10, 000 = 6,384,010,000 possible telephone numbers. That is an average
of almost 25 numbers per person!
A lock manufacturer can determine the number of possible combinations to
open its locks. If a dial lock has 60 numbers and requires three turns, then a total
of
60 × 60 × 60 = 216,000 locks can be made. However, some lock companies
do not want to have the same number listed twice, because dialing in different
directions might end up being confusing. Therefore it might be more appropriate
to create 
60 × 59 × 58 = 205,320 lock combinations. The 59 in the second posi-
tion means that there are 59 possible numbers available, because one number has
been selected in the first position; and the 58 in the third position indicates that
there are 58 possible numbers remaining, because one number has been selected
in the first position and a different number has been selected in the second posi-
tion. The product of three consecutive descending numbers is called a permuta-
tion. In this case, we would say that there are 60 permutations taken 3 at time,
meaning that the counting accounts for the selection of three numbers out of a
group of 60 in which the order of selection is important. Instead of writing the per-
mutation as a product of a series of integers
n(n − 1)(n − 2) • . . . • (n − r + 1),
it can be symbolized as nPr, where n is the number of possibilities for the first
selection, and r is the number of selections. 
Some counting principles are based on situations in which the order of selec-
tion is not important, such as in selecting winning lottery balls. If 6 numbers are
selected from a group of 40 numbers, it does not matter which number is pulled
out of the machine first or last. After all the numbers are randomly drawn, the
results are posted in numeric order, which is probably not the same order by
which they were selected. For example, if the numbers are drawn in the order 35–
20–3–36–22–28, and your ticket reads 3–20–22–28–35–36, then you are still the
winner. When order of selection is not important, this type of counting principle
is called a combination and can be symbolized as nCr. The relationship between 
a combination and permutation is determined by the equation nCr =
nPr
n!
because
there are n! ways to arrange a group of n objects, where n! = 
n(n − 1)(n − 2)
• . . . • 1. In this case, there are 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways to rear-
range 6 lottery balls with different numbers. Since the order of numbers is not
important when reading the winning lottery number, there are 
40
C
6
possible
numbers, or 
40×39×38×37×36×35
6×5×4×3×2×1
= 3, 838, 380 combinations, to select in the lot-
tery. In this type of lottery, the chance of winning would be 1 in 3,838,380.