Download Name: Section 11.1 – Permutations and Combinations Warm up

Name: _________________________
Section 11.1 – Permutations and Combinations
Warm up – How many different ways could you assemble a lunch consisting of a sandwich, a side, and a
dessert if a restaurant offers 4 kinds of sandwiches, 3 kinds of sides, and 4 desserts?
Fundamental Counting Principal - If event M can occur in m ways and is followed by event N that can
occur in n ways, then event M followed by event N can occur in m·n ways.
Example: 2 pants and 4 shirts give 2·4 possible outfits of pants and shirts.
Problem 1 – How many standard PA license plate possibilities are there if PA uses 3 letters followed by 4
numbers?
Problem 2 – How many more license plate possibilities would there be if we changed to a standard
license plate using just 7 letters?
Permutation – arrangement of items in a particular order. Supposed you wanted to arrange 3 items.
There are 3 ways you could choose the first item, 2 ways to choose second and 1 way to choose third.
Which is 3·2·1 = 6, or using factorial notation we write this 3!
Finding the number of permutations of n items:
Example 1 – In how many ways can you file 12 folders, one after another, in a drawer?
Example 2 – In how many ways can you arrange 8 shirts on hangers in a closet?
Number of Permutations – The number of permutations of n items of a set arranged r items at a time is:
nPr =
𝑛!
(𝑛−𝑟)!
for 0 ≤ r ≤ n
Example: 10P4 =
10!
=
(10−4)!
10!
6!
= 5040
Example 1 – Ten students are in a race. First through third will win a medal. In how many ways can 10
runners finish first, second, and third if no ties are allowed?
Method 1 - 10·9·8 = 720
Method 2 – nPr =
10!
(10−3)!
= 720
Example 2 – In how many ways can 21 runners finish first through fifth with no ties?
Method 1 –
Method 2 – nPr =
Example 3 – Fifteen students ask to visit a college admissions counselor. Each scheduled visit includes
one student. In how many ways can ten time slots be assigned?
Method 1 –
Method 2 – nPr =
Combinations – a selection in which order doesn’t matter.
Example – Ten students are in a race. First through third will advance to another final race. In this case
the order of the top three finishers doesn’t matter.
The number of combinations of n items of a set chosen r items at a time is…
nCr =
𝑛!
𝑟!(𝑛−𝑟)!
for 0 ≤ r ≤ n
Example - nCr =
5!
3!(5−3)!
=
5!
3!2!
=
120
6·2
= 10
Example 1 – What is 13C4, the number of combinations of 13 items taken 4 at a time?
*When determining whether to use permutations or combinations, we must determine whether order
is important or not.
Example – A chemistry teacher divides his class into eight groups. Each group will submit one group
project. He will then select four of the drawings to display. In how many different ways can he select
the drawings?
Example – You will draw winners from a total of 50 raffle tickets. The first ticket wins $100, the second
wins $50, and third wins $25. In how many different ways can you draw the three winning tickets?
Further practice: pg. 678 (39-41)