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Let's review what we already know about
the counting principle and permutations,
and introduce the term "combination".
(Additional review can be found at the
Algebra web site under Probability.)
Fundamental Principle of Counting:
(also known as the multiplication rule for
counting) If a task can be performed in n1 ways,
and for each of these a second task can be performed in n2
ways, and for each of the latter a third task can be performed in
n3 ways, ..., and for each of the latter a kth
task can be performed in nk ways, then the entire
sequence of k tasks can be performed in n1
• n2 • n3 • ... • nk
ways.
Permutation:
A set of objects in which position (or order) is
important.
To a permutation, the trio of
Brittany, Alan and Greg
is DIFFERENT from Greg,
Brittany and Alan. Permutations are
persnickety (picky).
Combination:
A set of objects in which position
(or order) is NOT important.
To a combination, the trio of
Brittany, Alan and Greg is THE SAME AS
Greg, Brittany and Alan.
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Let's look at which is which:
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Permutation
versus
Combination |
| 1.
Picking a team captain, pitcher, and shortstop from
a group. |
1. Picking three
team members from a group. |
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2. Picking your
favorite two colors, in order, from a color
brochure. |
2. Picking two
colors from a color brochure. |
| 3.
Picking first, second and third place winners. |
3.
Picking three winners. |
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Formulas:
| A
permutation
is the choice of r things from a set of n
things without replacement and where the order matters. |
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| A
combination
is the choice of r things from a set of n
things without replacement and where order does not
matter. (Notice the two forms of notation.) |
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The term
"combination" lock is mathematically confusing. To
open such a lock, the "order" of the digits entered IS
very important, unlike a mathematical combination.
New Name: Permutation
Lock |
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Example 1:
Evaluate  :
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Notice how the cancellation occurs, leaving
only 2 of the factorial terms
in the numerator. A pattern is emerging ... when finding a
combination
such as the one seen in this problem, the second value (2) will tell you
how many of the factorial terms to use in the numerator, and the
denominator will simply be the factorial of the second value (2). |
Example 2:
Joleen
is on a shopping spree. She buys six tops, three shorts and 4
pairs of sandals. How many different outfits consisting of a top, shorts
and sandals can she create from her new purchases?
(6)(3)(4) = 72
possible outfits
Example 3:
What is the total number of possible 4-letter arrangements of the
letters
m, a, t, h,
if each letter is used only once in each arrangement?
or
or
simply
4!
Example 4:
There are 12 boys and 14 girls in Mrs.
Schultzkie's math class. Find the number of ways Mrs. Schultzkie can
select a team of 3 students from the class to work on a group project. The team
is to consist of 1 girl and 2 boys.
Order, or position, is not important. Using the multiplication counting principle,
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How to use your
TI-83+/84+ graphing calculator with probability and
combinations.
Click calculator. |
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