MATH 30-1
Permutations, Combinations
& Binomial Theorem
Module Eight
Module / Unit 8 - Assignment Booklet
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Date Submitted: ___________________________________________
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Math 30-1: Module 8 Lesson Assignment
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Math 30-1: Module 8 Lesson Assignment
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Lesson 1: Fundamental Counting Principle
1. In Alberta, all postal codes start with the letter T. How many different postal codes are
possible in Alberta?
In Canada, all postal codes are made up of six characters, which follow the pattern letterdigit-letter-digit-letter-digit. The first letter of the code identifies a specific region. Each of
the remaining two letters in the postal code can be any of 20 letters, and each of the three
digits can be any digit from 0 to 9. All letters and digits can be repeated.
2. Find the number of possible unique positions for all five light switches.
Math 30-1: Module 8 Lesson Assignment
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3. Compared with the number of Alberta license plates available in 1912, find the increase in
the number of license plates available in 1941.
In 1912, Alberta license plates consisted of four digits. Each digit could be repeated, but
the first digit could not be zero. By 1941, Alberta license plates consisted of five digits.
Each digit could be repeated, but the first digit could not be zero.
4. Karen’s grandfather has a bank card that requires a four-digit Personal Identification
Number (PIN). All digits from 0 to 9 can be used.
a. How many PINs are possible if the digits can repeat?
b. How many PINs are possible if the digits cannot repeat?
5. Eight teenagers are lined up to buy tickets to a rock concert. How many possible
arrangements of teenagers are there? Leave your answer in factorial notation.
Math 30-1: Module 8 Lesson Assignment
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LESSON 1 SUMMARY
In this lesson you learned how to determine the total number of outcomes in a sample
space by using tree diagrams, outcome charts, and the fundamental counting principle.
You also learned a short-cut notation called factorial notation. By using both the
fundamental counting principle and factorial notation, you have seen that there are
many ways to solve problems that involve arranging items and finding the possible
number of arrangements.
The fundamental counting principle can be used to determine the number of possible
arrangements of items. If one task can be performed in a ways, a second task in b
ways, and a third task in c ways, the number of ways to calculate the possible
arrangements of all three tasks is a × b × c.
Factorial notation can be used as an abbreviation for products of successive
integers.
when n is a natural number.
Math 30-1: Module 8 Lesson Assignment
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Lesson 2: Permutations
1. Manny needs to create a four-digit password for his cell phone. If no digits can be
repeated, in how many ways can this task be done?
2. How many three-letter arrangements can be made from the letters in VERTICAL if no letter
can be used more than once and each arrangement is made up of a vowel between two
consonants?
3. In a gymnastics club, 10 gymnasts practice forming a human pyramid. The pyramid uses
only 6 gymnasts at any one time. There are 3 gymnasts in the bottom row, 2 in the second
row, and 1 at the top.
If any gymnast can fill any position in the pyramid, how many different arrangements of
gymnasts are there?
Math 30-1: Module 8 Lesson Assignment
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4. Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce,
1 poplar, 1 willow, and 1 elm. If the 4 birches are identical, how many possible
arrangements of trees are there?
5. Solve for r.
5Pr
= 60
6. Simplify
9!
.
6!
7. Solve the equation for n.
n ! = 42
n - 2!
8. Nine athletes have made it to the zone cross-country running finals. Based on her previous
times, Hariette is considered the favourite to win. If she does win, in how many different
orders can the other racers finish?
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Math 30-1: Module 8 Lesson Assignment
9. A local hockey club is setting up an online system to register its players.
Seven-character passwords will be assigned to each player, all starting with the letter S to
indicate the home club. The other characters in the password are made up of the numerals
4 through 9, and no digit can be repeated. How many passwords can the hockey club
create?
10. Explain why n must be greater than or equal to r in the expression
n Pr .
LESSON 2 SUMMARY
In this lesson you learned how to find the number of permutations of objects in several
ways. These ways included making a list, drawing a tree diagram, drawing blanks, using
the fundamental counting principle, and using the permutation formula, nPr. You learned
that there are fewer permutations if there are identical items and that some methods of
determining the number of permutations are better and more efficient than others
depending upon the question posed.
Permutations are used when the order of the objects is important.
nPr, where
0≤r≤n
The number of permutations of n objects when a objects are identical, b objects are
identical, c objects are identical, and so on is
In Lesson 3 you will learn about the number of choices of elements when order does not
matter.
Math 30-1: Module 8 Lesson Assignment
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Lesson 3: Combinations
A car dealership is promoting a particular model of car that has 8 optional features available.
Each optional feature can be purchased separately.
1. How many different packages of 3 optional features are possible for this model of the car?
2. Explain why the number of different packages of 5 optional features is the same as the
number of different packages of 3 optional features.
3. Another model of car has n different optional features available. When 2 optional features
are chosen for this model of car, there are 45 different packages available. Determine
algebraically the number of optional features, n, that are available for this model of car.
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4. How many different 5-card hands with at least 3 hearts can be dealt from a standard deck
of 52 cards?
Hint: See “Did You Know?” on page 535 of the textbook if you are not familiar with a
standard deck of cards.
5. How many 6-member committees can be formed from 8 girls and 9 boys if the following are
true?
a. There are no conditions.
b. There must be exactly 3 girls.
c. There must be at least 4 boys.
6. Create a question that involves both a permutation and a combination. Solve both showing
all your steps.
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LESSON 3 SUMMARY
Combinations can be used to solve counting problems where the order is not important.
A combination can be represented by nCr or
where n is the size of the group from
which the objects are taken and r is the number of objects that are taken at a time.
Many problems will require you to calculate multiple combinations and then apply the
fundamental counting principle.
Combinations are used when the order of the objects is not important. The following
formula can be used to calculate a combination where 0 ≤ n ≤ r
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Lesson 4: The Binomial Theorem
1. Write an explanation for a student in Mathematics 30-1 about how to expand (a  b)3.
Include at least two methods of expanding the power of the binomial.
2. Use the binomial theorem to expand (x2  2y)6.
æ
ö16
3. Determine the exact value of the 12th term in the expansion of ççx 2 - 1 ÷
.
÷
è
2ø
æ
ö8
4. Determine the term containing x7 in the expansion of ççx 2 + 1 ÷
.
÷
è
xø
LESSON 4 SUMMARY
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Expanding a binomial by multiplying is a very tedious process for any exponent larger
than 3. It is possible to use patterns to help simplify this process. The exponents of the
expansion follow an ascending and descending pattern, while the coefficients of the
expansion can be found using Pascal’s triangle.
Determining the values of Pascal’s triangle for large exponents would take a long time,
so combinations are used to determine entries of Pascal’s triangle to simplify this
process. The generalized result is the binomial theorem.
The binomial theorem is used to expand a binomial using a pattern that involves
combinations. The generalized binomial theorem is
Any term of the expansion can be determined using the formula
tk + 1 = nCk (x)n − k (y)k
Math 30-1: Module 8 Lesson Assignment
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MODULE 8 – PERMUTATIONS, COMBINATIONS & BINOMIAL
THEOREM SUMMATIVE ASSIGNMENT
Complete the following questions from your text book. Show steps completely and clearly, as
marks are assigned for mathematical literacy and communication. Always use graph paper,
rulers, and pencils as necessary. Attach securely to this booklet before you hand everything
in.
Text: Pre-Calculus 12 - Chapter 11
Section 11.1: Pages: 524 to 527 #2a,c, 6, 7a,c,d, 10
Section 11.2: Pages: 534 to 536 #3a, 5, 6a,c,d, 10
Section 11.3: Pages: 542 to 545 #2a,c, 3a,c, 5a, 6a, 7a,d,e, 12, 15, 17a,c, 18
Module 8 is now complete.
Once you have received your corrected work, review your instructor’s comments and prepare
for your module eight test.
Congratulations!!
Once all eight modules and module tests are completed, you are done the course!!! Now you
need to prepare for your Diploma Examination the date set provincially and for your final exam
to be written as soon as possible. Discuss these dates with Mrs. Martel!