1. 10.5
Permutations
and
Combinations

2. Permutations An arrangement of objects in which order is important
Example 1a: Find the number of permutations of the letters in the word TIGER.
5 options
4 options
3 options
2 options
1 option

3. How to express this as a formula? Let’s investigate!
Example 1b: In how many ways can you arrange 3 of the letters of the word TIGER?
How to express this as a formula? Let’s investigate!
5 options
4 options
3 options
Formula for permutations of
n objects taken r at a time

4. Circular Permutations
Five people are seated around a table:
In a row (linear permutation): A B C E D
ABCDE B A C D E
BCDEA C A B D E
CDEAB D A B C E
DEABC E A B C D
EABCD
In a circle (circular permutation):
There is only one circular permutation, but there are five corresponding linear ones.
For a set with n members,

5. Example 2
How many circular permutations are possible when seating five people around a table?
=24 permutations

6. Combinations An arrangement of objects in which order is NOT important
Example 2: Count the possible combinations of 2 letters chosen from the list A,B,C & D?
AB AC AD BA BC BD
CA CB CD DA DB DC
List
permutations
Remove repeats
There are 6 possible combinations.

7. Example 3: Count the possible combinations of 3 letters (out of 5) chosen from the word TIGER?
In Example 1b, we counted 60 different ways to arrange 3 letters chosen from the word TIGER when order is important.
Now, since order is NOT important, divide 60 by the number of ways to arrange the 3 letters that were chosen.
Number of arrangements for the 3 positions:
So,
Formula for
combinations:

8. Example 4:
If there are 9 students in a class, how many ways can a group of 1 to 3 students be formed?
Three separate choices for the number of students:
= 9 ways
1 student: 9C1=
= 36 ways
2 students: 9C2=
= 84 ways
3 students: 9C3=
Total:
= 129 ways

9. Probability ORDER IS NOT IMPORTANT
Example 5: From a standard deck of 52 cards, 5 cards are dealt. What is the probability that you receive only face cards?
ORDER IS
NOT
IMPORTANT

10. = = = = = Expansions of for small values of n Pascal’s Triangle 1 1 1

11. From Pascal’s
Triangle:
Investigate

12. Investigate =

13. Binomial Theorem
Example 6: Use Binomial Theorem to write the expansion of

14. Example 7: What is the coefficient of the
term in the expansion of ?
The term involves
So we want

15. Calculating Permutations and Combinations on TI-84
Enter value for n (e.g. [5])
For permutations, press [MATH], highlight PRB and select [2:nPr]
Enter value for r (e.g. [2])
Press [ENTER]
For combinations, follow same steps except select [3:nCr]

16. Calculating Permutations and Combinations on TI-89
For permutations
Press [2nd][MATH][7][2]
Enter value for n (e.g. [5])
Press [,]
Enter value for r (e.g. [2])
Press [ENTER]
For combinations, follow same steps except press [2nd][MATH][7][3]