Three points A, B, and C in three-dimensional
space are collinear if the triangle they form has zero
area. Equivalently, the three points are collinear if the
angle between the VECTORS
→
AB and
→
AC is zero, and
consequently the CROSS PRODUCT
→
AB ×
→
AC equals
the zero vector.
The collinearity of points in a plane is a topic of
interest to geometers. In the mid-1700s, LEONHARD
EULER discovered that several interesting points constructed from triangles are collinear, yielding his
famous EULER LINE. In 1893 British mathematician
James Sylvester (1814–97) posed the question of
whether it is possible to arrange three or more points in
a plane, not all on a line, so that any line connecting
two of the points from the collection passes through a
third point as well. Forty years later Tibor Gallai
(1912–92) proved that there is no such arrangement.
Two or more distinct PLANEs are said to be
collinear if they intersect in a common straight line. In
this case, the vectors normal to each plane all lie in a
plane perpendicular to the common line. Thus one can
determine whether or not a collection of planes is
collinear by noting whether or not the cross products
of pairs of normal vectors are all parallel.
See also GRADIENT; NORMAL TO A PLANE.
combination (selection, unordered arrangement) Any
set of items selected from a given set of items without
regard to their order is called a combination. Repetition
of choices is not permitted. For example, there are six
distinct combinations of two letters selected from the
sequence A,B,C,D, namely: AB, AC, AD, BC, BD, and
CD. (The selection BA, for example, is deemed the
same as AB, and the choice AA is not permitted.)
The number of combinations of k items selected
from a set of n distinct objects is denoted
. The
number
, for instance, equals six. The quantity
is called a combinatorial coefficient and is read as
“n choose k.” Given their appearance in the BINOMIAL
THEOREM, these numbers are also called binomial
coefficients.
One develops a formula for
by counting the
number of ways to arrange n distinct objects in a
row. There are, of course, n! different ways to do
this. (See FACTORIAL.) Alternatively, we can imagine
selecting which k objects are to be arranged in the
first k positions along the row (there are
ways
to do this), ordering those k items (there are k! different ways to do this), and then arranging the remaining n – k objects for the latter part of the row (there
are (n – k)! different ways to accomplish this). This
yields
different ways to arrange n objects
in a row. Since this quantity must equal n!, we have
the formula
for the combinatorial
coefficient.
It is appropriate to define 0! as equal to one. In
this way, the formula just established holds even for
k = n. (There is just one way to select n objects from
a collection of n items, and so
should equal
one.) It then follows that
. (There is just
one way to select no objects.) Mathematicians set
to be zero if k is negative or greater than n.
The combinatorial coefficients appear as the entries
of PASCAL’S TRIANGLE. They also satisfy a number of
identities. We list just four, which we shall phrase in
terms of the process of selecting k students to be in a
committee from a class of n students.
1.
(Selecting k students to be in a committee is the same as
selecting n – k students not to be in the committee.)
2.
(Any committee formed either includes, or excludes, a
particular student John, say. If John is to be on the
committee, then one must select k – 1 more students
n
k
n
k
n
k
=
−
−
+
−
1
1
1
n
k
n
n k
=
−
n
k
n
n
n
0
0
1
=
=
!
! !
n
n
n
n
=
!
! !
0
n
k
n
k n k
=
−
!
!(
)!
n
k
k n k
−
!(
)!
n
k
n
k
n
k
4
2
n
k
80 combination
space are collinear if the triangle they form has zero
area. Equivalently, the three points are collinear if the
angle between the VECTORS
→
AB and
→
AC is zero, and
consequently the CROSS PRODUCT
→
AB ×
→
AC equals
the zero vector.
The collinearity of points in a plane is a topic of
interest to geometers. In the mid-1700s, LEONHARD
EULER discovered that several interesting points constructed from triangles are collinear, yielding his
famous EULER LINE. In 1893 British mathematician
James Sylvester (1814–97) posed the question of
whether it is possible to arrange three or more points in
a plane, not all on a line, so that any line connecting
two of the points from the collection passes through a
third point as well. Forty years later Tibor Gallai
(1912–92) proved that there is no such arrangement.
Two or more distinct PLANEs are said to be
collinear if they intersect in a common straight line. In
this case, the vectors normal to each plane all lie in a
plane perpendicular to the common line. Thus one can
determine whether or not a collection of planes is
collinear by noting whether or not the cross products
of pairs of normal vectors are all parallel.
See also GRADIENT; NORMAL TO A PLANE.
combination (selection, unordered arrangement) Any
set of items selected from a given set of items without
regard to their order is called a combination. Repetition
of choices is not permitted. For example, there are six
distinct combinations of two letters selected from the
sequence A,B,C,D, namely: AB, AC, AD, BC, BD, and
CD. (The selection BA, for example, is deemed the
same as AB, and the choice AA is not permitted.)
The number of combinations of k items selected
from a set of n distinct objects is denoted
. The
number
, for instance, equals six. The quantity
is called a combinatorial coefficient and is read as
“n choose k.” Given their appearance in the BINOMIAL
THEOREM, these numbers are also called binomial
coefficients.
One develops a formula for
by counting the
number of ways to arrange n distinct objects in a
row. There are, of course, n! different ways to do
this. (See FACTORIAL.) Alternatively, we can imagine
selecting which k objects are to be arranged in the
first k positions along the row (there are
ways
to do this), ordering those k items (there are k! different ways to do this), and then arranging the remaining n – k objects for the latter part of the row (there
are (n – k)! different ways to accomplish this). This
yields
different ways to arrange n objects
in a row. Since this quantity must equal n!, we have
the formula
for the combinatorial
coefficient.
It is appropriate to define 0! as equal to one. In
this way, the formula just established holds even for
k = n. (There is just one way to select n objects from
a collection of n items, and so
should equal
one.) It then follows that
. (There is just
one way to select no objects.) Mathematicians set
to be zero if k is negative or greater than n.
The combinatorial coefficients appear as the entries
of PASCAL’S TRIANGLE. They also satisfy a number of
identities. We list just four, which we shall phrase in
terms of the process of selecting k students to be in a
committee from a class of n students.
1.
(Selecting k students to be in a committee is the same as
selecting n – k students not to be in the committee.)
2.
(Any committee formed either includes, or excludes, a
particular student John, say. If John is to be on the
committee, then one must select k – 1 more students
n
k
n
k
n
k
=
−
−
+
−
1
1
1
n
k
n
n k
=
−
n
k
n
n
n
0
0
1
=
=
!
! !
n
n
n
n
=
!
! !
0
n
k
n
k n k
=
−
!
!(
)!
n
k
k n k
−
!(
)!
n
k
n
k
n
k
4
2
n
k
80 combination