perspective 391

To prove this, let Si1i2…ikdenote the number of per-

mutations that keep the elements i1,i2,…ikfixed in

place. (Thus the remaining n– kobjects may be per-

muted in any fashion.) We have Si1i2…ik = (n– k)!.

There are ways to choose kobjects, and by the

INCLUSION

-

EXCLUSION PRINCIPLE

there are thus

permutations that keep at least one object in its correct

location. The formula above follows readily.

The

PROBABILITY

that a permutation chosen at ran-

dom is a derangement is this number divided by n!, the

total number of permutations. If nis large, this is very

close in value to the infinite series:

which happens to be the T

AYLOR SERIES

of exevaluated

at x= – 1. This solves the famous “hatcheck problem”:

A group of gentlemen check their hats at a

country-club cloakroom. The clerk loses

records of which hat belongs to whom and

starts handing out hats randomly as the men

leave. What is the probability that no man

receives his own hat?

The answer is approximately .

Permutations are also used to analyze the

SLIDE FIF

-

TEEN PUZZLE

. The number of permutations of nobjects

in which each object moves at most one place, either

left or right, from its original location is Fn+1, the

(n+1)-th F

IBONACCI NUMBER

.

See also

BINOMIAL COEFFICIENT

;

COMBINATION

;

FUNCTION

;

SUBFACTORIAL

.

perpendicular This term is used in any setting to

describe two geometric constructs that meet at right

angles. For example, two lines are perpendicular if the

angle between them is 90°. To “drop a perpendicular”

from a point to line is to draw a line segment that

starts at the given point and meets the line at right

angles. The length of this line segment is the shortest

distance between the given point and points on the line.

This length is called the perpendicular distance of the

point from the line.

The perpendicular bisector of a line segment is the

line at right angles to the segment passing through its

midpoint. It represents all points in the plane that are

equidistant from the two endpoints of the given line

segment. Two planes are perpendicular if they meet at

right angles.

The term perpendicular tends to be used primarily

in discussions about lines and planes. Other geometric

quantities (such as

VECTOR

s and curved surfaces) might

meet at right angles, but mathematicians tend to use

the word

ORTHOGONAL

in these settings. For example,

two vectors are orthogonal if the angle between them is

90°. This convention is not steadfast. Often the words

perpendicular and orthogonal are used interchangeably.

See also

DOT PRODUCT

;

NORMAL TO A CURVE

;

NOR

-

MAL TO A PLANE

;

NORMAL TO A SURFACE

;

SLOPE

.

perspective Two planar figures are said to be in per-

spective from a point Pif, for each point Aon one fig-

ure, there is a corresponding point Bon the other so

that the line connecting Pto Aalso meets B. The point

Pis called the center of perspective.

The notion of perspective played a significant role

in the development of artistic techniques in the 15th

century. In an attempt to capture a sense of depth in a

two-dimensional painted scene, Renaissance artists

began drawing objects in the foreground larger than

those of the same size in the background. They imag-

ined a distant point at infinity at the back of the scene

and drew guidelines emanating from this point across

the canvas to aide in creating three-dimensional realism.

(Today we call such a construct a central

PROJECTION

.)

Artists and scholars A

LBRECHT

D

ÜRER

(1471–1528)

and G

IRARD

D

ESARGUES

(1591–1661) were the first to

study the mathematics of perspective.

Mathematicians also say that two planar figures

are in perspective from a line if corresponding sides of

each shape, if extended, meet at points that all lie on a

common line. D

ESARGUES

’

S THEOREM

shows that if two