PreCalculus 2 Notes 13-2: Day 1
Combinations, permutations, and the multiplication principle of counting
A permutation is an ordered arrangement of r objects chosen from n objects.
3 types of permutations
Order is important!
Think-Lock Combination!
Type 1: n objects are distinct (different) and repetition is allowed in the selection of r of them.
(Distinct, with repetition)
Example:
We want to establish a two letter codes can be formed using the letters of A,B,C? Repeated letters are
allowed.
Example:
The international Airline Transportation Association assigns three letter codes to represent airport locations.
For example, ORD is for O’Hara airport in Chicago. FLL is the airport code for Ft. Lauderdale, Florida. EVV is for
Evansville Indiana. Notice that repetition is allowed in forming this code. How many airport codes are
possible?
Example:
How many three-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? Repeated digits are
allowed.
The number of ordered arrangements of r objects chosen from n objects, in which the n objects are distinct
and repetition is allowed, is n r
Type 2: The n objects are distinct(different), and repetition is not allowed in the selection
of r of them, where r  n.
(distinct, without repetition)
Example:
We want to establish a two letter code using the letters of A,B,C but we require that no letter be used more
than once. How many different two-letter codes are there?
The notation P (n, r ) represents the number of ordered arrangements of r objects chosen from n distinct
objects, where r  n and repetition is not allowed.
Permutations of r Objects Chosen from n Distinct Objects without Repetition:
The number of arrangements of n objects using r  n of them, in which
1. The n objects are distinct
2. Once an object is used it cannot be repeated
3. Order is important!
Formula: The number of permutations of n objects taken r at a time is denoted n Pr or P(n, r ) where
n Pr 
n!
(n  r )!
n Factorial  n
Example: 5!=
Example:
Example:
 n(n  1)    3  2  1
5, then Calculator MATH over to PRB #4
3!
=
 3  2 !
P 
3 2
3, then Calculator MATH over to PRB #2, then 2
Example:
How many different four-letter codes are there if only the letters A, B, C, D, E and F can be used and no letter
can be used more than once?
Example: List all the ordered arrangements of 6 objects 1,2,3,4,5, and 6 choosing 3 at a time without
repetition?
Example: In how many ways can 4 people be lined up?
Example: Suppose that we wish to establish a three letter code using any of the 26 uppercase letters of the
alphabet, but we require that no latter be used more than once. How many different three letter codes are
there?
Example: We know 3 people in the class have different birthdays. If we listed all the possible ways this could
occur, how many would there be? Assume that there are 365 days in a year.
Combinations (order doesn’t matter)
Making a Smoothie!
Making a Salad!
The notation C(n, r ) represents the number of combinations of n distinct objects using r of them.
The number of combinations of n objects selected r at a time is denoted n Cr
The number of arrangements of n objects using r  n of them , in which
1. The n objects are distinct
2. Once an object is used, it cannot be repeated
3. Order is not important
Formula:
n Cr 
n!
(n  r )! r !
Example:
List all the combinations of 5 objects a,b,c,d and e taken 2 at a time. List them.
That is, Find: 5 C2
Example: In how many different committees of 3 professors can be formed from a department having 8
professors?
Example: In how many ways can a committee of 5 students be formed from a pool of 15 students?
Give a problem describing each and write out the solution:
4
C2
4
P2