Greenwood press

20
CONIC SECTIONS

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How to build a parabolic reflector

Elliptical orbits


Elliptical gears


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COUNTING
Businesses and government agencies often have a need to efficiently count
the number of arrangements or possibilities with various combinations of num-
bers or options. For example, a car dealer may be interested in the number of car
varieties that can be offered in order to persuade customers. If there are 9 differ-
ent models, 6 different colors, and 2 types of interiors, there could be a total of 
9
× 6 × 2, or 108, different cars available. In this dealer’s television advertise-
ment you might hear, “Hurry, this weekend only. Come to our car dealership and
view over 100 different styles of cars for sale. Don’t miss out on this great oppor-
tunity!” The procedure of multiplying the number of possibilities for each option
is called the multiplication counting principle.
State vehicle departments can determine the number of nonvanity license
plates they have available by finding the product of the number of possibilities
for each position on the plate. For example, if a state has three letters followed
by three numbers, then the number of possible plates is 9 
× 10 × 10 × 26 × 26
× 26 = 9 × 10
2
× 26
3
= 15,818,400. The first position will have 9 possible val-
ues, since it will represent any digit from 1 through 9. The second and third posi-
tions can hold 10 possible digits from 0 to 9. The fourth through sixth positions
have 26 possibilities each, because they can contain any letter in the alphabet. If
for some reason the state runs out of license-plate sequences, they can place
numbers before letters to double the number of possibilities, since the order of
letters and numbers is important on a license plate. Zip codes can be counted in
a similar manner. There are five numbers in a zip code, so there is a total of
10 × 10 × 10 × 10 × 10 = 10
5
= 100,000 possible zip codes. The United States
Postal Service only uses 95,000 possibilities: 00001 to 95000.
Telephone numbers are counted in a similar way, but have more restrictions
to the values in different positions. The first three digits are the area code. The
area code must start with a digit from 2 to 9, because pressing 0 is a call to the
operator, and pressing 1 is not allowed because it is associated with dialing a
number outside an area code. Also, the area code cannot be 911, since that is an