Permutations and Combinations
MDM 4U: Mathematics of Data Management
Unit: Counting and Probability
By: Mr. Allison and Mr. Panchbhaya
Specific Expectations
Strand 2.1
• Recognize the use of permutations and
combinations as counting techniques with
advantages over other counting techniques
Strand 2.2
• Solve simple problems using techniques for
counting permutations and combinations,
where all objects are distinct
Learning Goals
• Make connections between, and learn to
calculate various permutations and
combinations
• Learn to behave in class
Agenda of the Day
•
•
•
•
Probability Video
Review
Worksheet
Game show Activity
How many combinations would it take for
the tire to attach itself back to the car?
Real Life Examples
• Video game designers
– to assign appropriate scoring values
• Engineering
– new products tested rigorously to determine how
well they work
• Allotting numbers for:
– Credit card numbers
– Cell phone numbers
– Car plate numbers
– Lottery
Factorials
• The product of all positive integers less than
equal or equal to n
n! = n x (n – 1) x (n – 2) x … x 2 x 1
5! =5 x 4 x 3 x 2 x 1 = 120
Permutations
• Ordered arrangement of objects selected from a
set
• Ordered arrangement containing a identical objects
of one kind is
𝑛!
𝑎!𝑏!𝑐!…
Combinations
• Collection of chosen objects for which order
does not matter
Speed Round: The sports apparel store at
the mall is having a sale. Each customer
may choose exactly two items from the list,
and purchase them both. The trick is that
each 2-item special must have two
different items (for example, they may not
purchase two T-shirts at the same time).
What are all the different combinations
that can be made by choosing exactly two
items?
15 combinations are possible
Q – How many combinations are made
if you were purchasing three items
instead of two?
1. A club of 15 members choose a
president, a secretary, and a treasurer in
A. 455 ways
B. 6 ways
C. 2730 ways
2. The number of debate teams
formed of 6 students out of 10 is:
A. 151200
B. 210
C. 720
3. A student has to answer 6 questions
out of 12 in an exam. The first two
questions are obligatory. The student
has:
A. 5040
B. 210
C. 720
4. From a group of 7 men and 6 women,
five persons are to be selected to form a
committee so that at least 3 men are
there on the committee. In how many
ways can it be done.
A.
B.
C.
D.
E.
564
645
735
756
None of the above
5. In how many different ways can the
letters of the word “LEADING” be
arranged in such a way that the vowels
A.
B.
C.
D.
E.
360
480
720
5040
None of the above
6. How many permutations of 4
different letters are there, chosen from
the twenty six letters of the alphabet
(repetition is not allowed)?
Answer
The number of permutations of 4 digits chosen
from 26 is 26P4 = 26 × 25 × 24 × 23 = 358,800
How many paths are there to the top of the board?
Answer
How many 4 digit numbers can be made using 0-7 with
no repeated digits allowed?
a)
b)
c)
d)
5040
4536
2688
1470
Answer
• = 7x7x6x5 = 1470
• First digit of a number can not be ‘0’
No postal code in Canada can begin with the letters D,F,I,O,Q,U, but repeated
letters are allowed and any digit is allowed. How many postal codes are
possible in Canada?
a)
b)
c)
d)
11,657,890
13,520,000
14,280,000
12,240,000
Answer
• = 20x10x26x10x26x10 = 13,520,000
• 20 choices for the first letter (26 - 6 that cannot be chosen.
10 choices for the digit (0-9).
• 26 choices for the 3 position (2nd letter)
• then 10 choice for the 4th position
• Then 26 and 10 since you can again repeat numbers and
letters.
Using digits 0 – 9, how many 4 digit
numbers are evenly divisible by 5 with
repeated digits allowed?
a)
b)
c)
d)
1400
1600
1800
1500
Answer
• 9 × 10 × 10 × 2 = 1800
• First # can’t be ‘0’
• Last # has to be ‘5’ or ‘0’
How many ways can you arrange the letters in
the word REDCOATS if it must start with a vowel
a)
b)
c)
d)
15,120
14,840
15,620
40,320
Answer
• 3* × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 15,120
• EOA are your 3 choices
How many groups of 3 toys can a child choose to
take on a vacation from a toy box containing 11
toys?
a) 990
b) 1331
c) 165
d) 286
Answer
• C(11,3) =
11!
8! 3!
= 165
If you have a standard deck of cards how
many different hands exists of 5 cards
a)
b)
c)
d)
2,598,960
3,819,816
270,725
311,875,200
Answer
• C(52,5) =
52!
47! 5!
= 2,598,960
The game of euchre uses only 24 cards from a standard
deck. How many different 5 card euchre hands are
possible?
a)
b)
c)
d)
7,962,624
42,504
5,100,480
98,280
Answer
• C(24,5) =
24!
19! 5!
= 42,504
Solve for n 3(nP4) =n-1P5
a)
b)
c)
d)
8
10
2
5
Answer
How many ways can 3 girls and three boys sit in
a row if boys and girls must alternate?
Answer
• = 3! x 3! + 3! x 3!
•
3 3 2 2 1 1
𝑥 𝑥 𝑥 𝑥 𝑥
𝐵 𝐺 𝐵 𝐺 𝐵 𝐺
• = 72
+
3 3 2 2 1 1
𝑥 𝑥 𝑥 𝑥 𝑥
𝐺 𝐵 𝐺 𝐵 𝐺 𝐵
Laura has ‘lost’ Jordan’s phone number. All she
can remember is that it did not contain a
0 or 1 in the first three digits. How many 7 digit
#’s are possible
Answer
• = 8 x 8 x 8 x 10 x 10 x 10 x 10
• = 5,120,000