Chung

Tip 13

Permutation & Combination

• Permutation is an arrangement of objects in some specific order. A selection in which order does not matter is called a combination

– The number of permutations of n things taken n at a time is

• 𝑃𝑛 𝑛 = 𝑛 𝑛 − 1 𝑛 − 2 𝑛 − 3 ∙∙∙ 2 ∙ 1 = 𝑛!

– Example: How many ways can 5 people, standing in line, be arranged?

• 𝑃5 5 = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 5! = 120

• Permutations – The number of permutations of n things taken r at

a time is

𝑃𝑛 𝑟 =𝑛!

(𝑛−𝑟)!

• Combinations – The number of combinations of n things taken r at

a time is

𝐶𝑛 𝑟 =𝑛!

𝑛−𝑟 !𝑟!

ORDER IS IMPORTANT!

ORDER IS NOT IMPORTANT!

Permutations

• How many different ways are there to rearrange the letters in APRIL?

• How may ways can you arrange any three of the letters in APRIL?

Permutations With Repetition

• How many ways can you rearrange the letters in TWITTER? In TWEET? In TOOT?

Circular Permutations

• A club has four officers, President; Vice President; Secretary; & Treasurer. How many different ways can these 4 be seated at a round table? President

Vice President

Treasurer

Secretary

Handshakes

• If there are 5 people in a room and they shake each other’s hand only once, how many handshakes are there altogether?

• If you have 12 people in a group and each person shakes hands only once with every person, how many handshakes?

Warm-Up 10/15

• If there are 5 lines in a plane, what is the greatest number of possible intersections?

• Five points lie on a circle. If a line segment is formed between any 2 points, what is the greatest number of line segments?

Warm-up 10/18

• How many ways can two boys be selected from 11 boys?

Percent of a solution

• The percent of a solution is expressed as the percentage of solute over the total amount of solution. 𝑝% of a solution is:

𝑆𝑜𝑙𝑢𝑡𝑒

𝑇𝑜𝑡𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛=

𝑝

100

• How many gallons of water must be added to 40 gallons of a 10% alcohol solution to produce a 8% alcohol solution?

40 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 ×10%

𝑥 + 40 𝑔𝑎𝑙𝑙𝑜𝑛𝑠=

8

100

• How many gallons of a 20% salt solution must be added to 10 gallons of a 50% salt solution to produce a 30% salt solution?

.20𝑥+5

𝑥+10=

30

100

Tip 16: Slope of a Line

• One of the most important properties of a straight line is its angle from the horizontal. This concept is called “slope”. To find the slope, we need two points from the line.

• From two points 𝑥1, 𝑦1 𝑎𝑛𝑑 𝑥2, 𝑦2

• Slope 𝑚 =𝑦2−𝑦1

𝑥2−𝑥1

• From slope – intercept form of a line • 𝑦 = 𝑚𝑥 + 𝑏, where m is slope & b is y-intercept.

• In the figure shown, a point P(42, t) lies on the line. What is the value of t?

A. 39

B. 42

C. 45

D. 52

E. 60

• If 𝑓 is a linear function and 𝑓 3 = 6 𝑎𝑛𝑑 𝑓 5 = 12, what is the slope of the graph of 𝑓?

A. 2

B. 3

C. -2

D. -3

E. -4

• Three points on a line are (2,5), (4,a), and (8,23). What is the value of a?

A. 11

B. 14

C. 15

D. 16

E. 18

Parallel vs. Perpendicular lines

• Parallel lines have identical slopes.

• Perpendicular slopes are negative reciprocals of each other.

• To find the equation of a line through a given point 𝑃(𝑥1, 𝑦1), use the appropriate slope in the point/slope formula: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Examples

• A line 𝐿1 that passes through the point (0, -2) is

perpendicular to line 𝐿2 with a slope of 1

2.

• What is the equation of the 1st line?

𝑦 − −2 = −2 𝑥 − 0

𝑦 + 2 = −2𝑥 -2 because negative

reciprocal of 1

2

Tip 17 – Number of Factors

• Let n be a natural number with prime factorization 𝑛 = 𝑎𝑘1𝑏𝑘2𝑐𝑘3. The number of factors of the number is 𝑘1 + 1 𝑘2 + 1 𝑘3 + 1 .

– The number of factors (positive divisors) can be found by adding one to all exponents of prime factors and multiplying those results together.

– Example: For a natural number 12, 12 = 22 × 31. The number of factors is (2 + 1)(1 + 1) = 6

• Let a positive integer be defined by 𝑛 = 𝑝2 ×𝑞4, where p and q are distinct prime numbers. How many factors does the number n have?

A. 6

B. 8

C. 12

D. 15

E. 20

Warm-up

• Let a positive integer k be defined by 𝑘 = 24𝑝3, where p is a prime number greater than 5. How many factors does the number k have? A. 8

B. 16

C. 24

D. 32

E. Cannot be determined

Tip 18: Composition of Functions

• Composition of functions is applying one function to the result of another function.

• The result of 𝑓 is sent through 𝑔( )

• The composition of functions is written in the

form 𝑔 𝑓 𝑥 𝑜𝑟 (𝑔 ∘ 𝑓)(𝑥)

𝑓( ) 𝑔( )

Composition: Examples

• Given the function 𝑓 𝑥 = 3𝑥 𝑎𝑛𝑑 𝑔 𝑥 =𝑥 − 4, 𝑡ℎ𝑒𝑛 𝑔(𝑓 3 ) =

• For the given functions 𝑓 𝑥 = 3𝑥 −4 𝑎𝑛𝑑 𝑔 𝑥 = 2𝑓 𝑥 + 3, 𝑖𝑓 𝑔 𝑘 = 0,what is the value of k?

𝑓 = 3𝑥 𝑔 = 𝑥 − 4 𝑥 = 3 9 9 – 4 = 5

Warm-up 10/23

• The function g is defined by 𝑔 𝑥 = 3𝑓 𝑥 −𝑘, and the function f is defined by 𝑓 𝑥 =5𝑥 + 3. 𝐼𝑓 𝑔 2 = 25, what is the value of k?

A. 18

B. 14

C. 12

D. 10

E. 8

Tip 19: Consecutive Integers

• What is the sum of 11 consecutive integers if the middle one is 30?

A. 60

B. 120

C. 330

D. 660

E. 990

Tip 19: Consecutive Integers

• Integers which follow each other in order, without gaps, from smallest to largest are consecutive integers

• Properties – For consecutive integers, if the 1st term is 𝑎1 and

the last term is 𝑎𝑛, the average = the median

• Average (Arithmetic mean) =𝑎1+𝑎𝑛

2

• Sum of consecutive = median x number of integers, or

• Sum of consecutive = average x number of integers

• If the median of a list of 99 consecutive integers is 80, what is the greatest integer in the list?

A. 99

B. 128

C. 129

D. 157

E. 179

Since the 50th number is 80, the 99th number is 80 + (99 – 50) = 129

• The median of a list of 10 consecutive even integers is 77. What is the sum of the integers?

A. 700

B. 770

C. 780

D. 800

E. 870

• If the median of a list of 30 consecutive odd integers is 120, what is the greatest integer in the list?

A. 145

B. 147

C. 149

D. 151

E. 167

Tip 20 - Must be True – Could be True

• (𝑎 + 𝑏)2= (𝑎 − 𝑏)2 The question might be as follows:

1) If the statement above is true, which of the following must also be true (always true)? or

2) If the statement above is true, which of the following could be true (possibly true)

• If 𝑎 > 𝑏, 𝑎𝑛𝑑 𝑎 𝑎 − 𝑏 = 0, which of the following must be true?

I. 𝑎 = 0

II. 𝑏 < 0

III. 𝑎 + 𝑏 < 0

A. I ONLY

B. II only

C. III only

D. I and II only

E. I and II and III only

• Because 𝑎 − 𝑏 ≠ 0, 𝑎 must be 0.

• From the given 𝑎 > 𝑏, because 𝑎 = 0, 𝑏 < 0 “must be true”.

• Since 𝑎 must be 0 and 𝑏 < 0 “must be true”, 𝑎 + 𝑏 < 0 must be true.

• For real numbers 𝑎 𝑎𝑛𝑑 𝑏, 𝑎 − 𝑏 = 𝑎 + 𝑏, where 𝑎 > 𝑏. Which of the following must be true?

I. 𝑏 = 0

II. 𝑎 > 0

III. 𝑎 = 0 A. I only

B. II only

C. III only

D. I and II only

E. I and III only

• Square both sides 𝑎 − 𝑏 = 𝑎 + 𝑏 → 𝑏 = 0

Since 𝑎 > 𝑏, 𝑎 > 0

• If 𝑎2 + 𝑏2 = 𝑎 − 𝑏,𝑤ℎ𝑒𝑟𝑒 𝑎 is positive, which of the following must be true?

A. 𝑎 = 0

B. 𝑏 = 0

C. 𝑎𝑏 > 0

D. 𝑎 = 1

E. 𝑎 − 𝑏 = 1

• 𝑎2 + 𝑏2 = 𝑎2 − 2𝑎𝑏 + 𝑏2

• 2𝑎𝑏 = 0 → 𝑎𝑏 = 0

• Since a is positive, b must be 0.

Tip 21 - Sum and the Number of Consecutive Integers

• The smallest integer of a set of consecutive integers is – 10. If the sum of these integers is 23, how many integers are in this set?

• −10 … … + 10,11,12 We know the sum of the consecutive integers between – 10 and + 10 is zero. 10 integers before zero, plus zero, plus 10 integers after zero = 21 integers. Two more integers (11 and 12) are needed to sum to 23. Therefore, 23 integers are in the set.

• If the sum of the consecutive integers from – 30 to x, inclusive, is 96, what is the value of x?

A. 30

B. 31

C. 32

D. 33

E. 34

• The smallest integer of a set of even consecutive integers sum is – 20. If the sum of these integers is 72, how many integers are in the set? A. 24

B. 25

C. 43

D. 44

E. 45

• The greatest integer in a set of consecutive integers is 61. If the sum of these integers is 61, how many integers are in this set?

– 2

– 61

– 121

– 122

– 125

Tip 22 – No Solution

• A system of linear equations means two or more linear equations. – The system has exactly one solution

• When two lines have different slopes

– The system has no solution • When two lines are parallel and have different y-

intercepts

– The system has infinitely many solutions • When two lines are parallel and have the same y-

intercept

𝒂𝟏𝒙 + 𝒃𝟏𝒚 = 𝒄𝟏 and 𝒂𝟐𝒙 + 𝒃𝟐𝒚 = 𝒄𝟐

• If 𝑎1

𝑎2≠

𝑏1

𝑏2 One

• If 𝑎1

𝑎2=

𝑏1

𝑏2≠

𝑐1

𝑐2 None

• If 𝑎1

𝑎2=

𝑏1

𝑏2=

𝑐1

𝑐2 Infinite

𝒚 = 𝒎𝟏𝒙 + 𝒃𝟏 and 𝒚 = 𝒎𝟐𝒙 + 𝒃𝟐

• If 𝑚1 ≠ 𝑚2

• If 𝑚1 = 𝑚2 𝑎𝑛𝑑 𝑏1 ≠ 𝑏2

• If 𝑚1 = 𝑚2 𝑎𝑛𝑑 𝑏1 = 𝑏2

Standard Form Slope – Intercept Form

2𝑥 − 5𝑦 = 8 4𝑥 + 𝑘𝑦 = 17

• For which of the following values of k will the system of equations above have NO solution?

A. 10

B. 5

C. 0

D. - 5

E. - 10

2

4=

−5

𝑘≠

8

17

5𝑥 − 2𝑦 = 3 𝑎𝑥 + 𝑏𝑦 = 6

• For the system of equations above, the system has infinite solutions. What is the value of 𝑎 + 𝑏?

A. 6

B. 4

C. 0

D. - 4

E. - 6

5

𝑎=

−2

𝑏=

3

6

3𝑥 + 𝑏𝑦 = 3 𝑎𝑥 − 4𝑦 = 6

• For which of the values of 𝑎, 𝑏 will the system of equations have NO solution?

A. −1,2

B. 1,1

C. 2,1

D. 3, −4

E. 6,2

3

𝑎=

𝑏

−4≠

3

6