Lauretta J. Fox
A
combination
is a group of objects in which the order or arrangement of the objects is disregarded. From the numbers 1, 2, 3, six different permutations can be formed. They are 123, 132, 231, 213, 312, 321. When the order of the digits is not considered, all six of these permutations make up one combination. The number of combinations of three objects, taken three at a time is one. 303 =
1. In general, the number of combinations of n objects taken n at a time is one. nCn =
1. When the number of combinations of n objects, taken n at a time, is multiplied by n!, the result equals nPn.
____
____
____
nPn
|
nC n x n! = nP n
|
or
|
nC n= n;
|
Similarly, the number of combinations that can be formed from n objects, taken r at a time is
|
nC r = nP y =
|
|
n!
|
|
r! =
|
|
(n—r )! r!
|
The number of combinations of n things, taken r at a time, is equal to the number of combinations of n things, taken n-r at a time, that is, nC r = nC nÐr.
nC nÐr = n! = n! = nC r
|
[n-(n-=r)] ! (n-r)!
|
r! (n-r)!
|
Specifically, 40
C
38 = 40
C
2 = 40! = 40x39x38! = 780
In many cases this fact can be used to simplify computations.
Example 1
: In how many ways can a committee of four be chosen from ten people?
Solution: 10C
4 = 10! = 10 x 9 x 8 x 7 x 6! = 210
(10-4): 4! 6! x 4 x 3 x 2
Example 2:
How many combinations can be made from seven objects, using them five at a time?
Solution:
7
C
5 = 7! = 7 x 6 x 5! = 42 = 21
Example 3:
Evaluate 60
C
57.
Solution:
60
C
57 = 60
C
3 = 60! = 60!
|
|
|
(60-3)!
|
3! 57! 3!
|
|
|
= 60 x 59 x 58 x 57! = 34,220
|
|
|
57! x 1 x 2 x 3
|
Exercises
-
1.) Find the number of combinations of five objects taken from a group of nine objects.
-
2.) How many combinations of four items are there in a given set of six items?
-
3.) How many diagonals can be drawn in an octagon?
-
4.) In how many ways can seven questions out of ten be chosen on an examination?
-
5.) In how many ways can three books be chosen from five books?
-
6.) From a group of twelve ladies a committee of three is to be selected. In how many ways can this committee be formed with Mrs. Adams on the committee, but with Mrs. Jones excluded, if these two are part of the group of twelve?
-
7.) How many committees can be formed from a group of eight men, if one particular member of the group is to be included and two other members of the group are to be excluded from the committee?
