USAD Math
General Mathematics, Geometry
Introduction to Differential Calculus
General Mathematics
Simple and Compound Interest
Permutation and Combination
Probability
Geometry
Right Triangle
Coordinate Geometry
Plane and Solid Figures
Circles
Differential Calculus
Polynomial
Limits and Continuity
Derivative
General Mathematics
Interests
Single Interest
The principal balance do not change
Single Interest
Formula
A=P(1+rt)
A
Final Value
P
Principal balance
Rate
r
t
Month, Year
Example
Borrow $500 from the bank with simple interest 5% per week
How much do you need to pay back after a month?
Example
Borrow $500 from the bank with simple interest 5% per week
How much do you need to pay back after a month?
500 + 4 (500\times0.05) = 600
Compound Interest
The amount of interest earned would be added
in the principal amount of money
For example
For example
You save $10000 in a bank with a rate 1%
compounded annually
How much money would you get after 10 years?
For example
You save $10000 in a bank with a rate 1% compounded annually
How much money would you get after 10 years?
Solution
10000(1 + 1 \%) = 10100 \rightarrow 10100(1+1\%) = 10201
do the upper calculation 10 times
10000(1+1\%)^{10} = 11046
Compound Interest
Formula
A=P(1+\frac{r}{n})^{nt}
A
Final Value
P
Principal balance
Rate
(Monthly, Annual)
r
t
Month, Year
number of
interests applied in a time period
n
For example
You save $10000 in a bank with a rate 1%
compounded monthly
How much money would you get after 10 years?
For example
You save $10000 in a bank with a rate 1% compounded monthly
How much money would you get after 10 years?
Solution
10000(1 + \frac{1 \%}{12}) = 10008 \rightarrow 10100(1+\frac{1\%}{12}) = 10016
do the upper calculation 10 times
10000(1+\frac{1\%}{12})^{10\times12} = 11051
Compound Continuously
The value compound continuously
Compound Continuously
The value compound continuously
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}
Type
in your calculator
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}
Type
in your calculator
We will get a value
2.718...
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{r}{n})^{nt}}
\displaystyle{\lim_{n\to\infty}(1+\frac{1}{n})^{n}}
Type
in your calculator
We will get a value
2.718...
This number, we use a symbol to represent it
e
\displaystyle{A = P\lim_{n\to\infty}(1+\frac{1}{n})^{nt}}
We rewrite this formula
A = Pe^{rt}
For example
You save $10000 in a bank with a rate 1%
compounded continuously
How much money would you get after 10 years?
For example
You save $10000 in a bank with a rate 1% compounded continously
How much money would you get after 10 years?
Solution
10000e^{0.01\times10} = 11051
Permuation & Combination
What does these two word means?
What does these two word means?
Four people are waiting in a row, how many possible ways can they form?
We want to choose three people from a class
to participate in a contest
How many ways can we choose?
What does these two word means?
Four people are waiting in a row, how many possible ways can they form?
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
P
C
Four people are waiting in a row, how many possible ways can they form?
| ABCD | BACD | CABD | DABC |
|---|---|---|---|
| ABDC | BADC | CADB | DACB |
| ACBD | BCAD | CBAD | DBAC |
| ACDB | BCDA | CBDA | DBCA |
| ADBC | BDAC | CDAB | DCAB |
| ADCB | BDCA | CDBA | DCBA |
Four people are waiting in a row, how many possible ways can they form?
| ABCD | BACD | CABD | DABC |
|---|---|---|---|
| ABDC | BADC | CADB | DACB |
| ACBD | BCAD | CBAD | DBAC |
| ACDB | BCDA | CBDA | DBCA |
| ADBC | BDAC | CDAB | DCAB |
| ADCB | BDCA | CDBA | DCBA |
Four people are waiting in a row, how many possible ways can they form?
24
Four people are waiting in a row, how many possible ways can they form?
A
B
C
D
B
C
D
C
D
D
4\times3\times2\times1
24
n people are waiting in a row, how many possible ways can they form?
n people are waiting in a row, how many possible ways can they form?
n\times(n-1)\times(n-2)\times...\times1
n people are waiting in a row, how many possible ways can they form?
n\times(n-1)\times(n-2)\times...\times1
n!
We define ! as the symbol of the calculation
We call it "Factorial"
And define
0! = 1
There are 6 people and 3 seats
How many ways can they form?
There are 6 people and 3 seats
How many ways can they form?
6\times5\times4
There are 6 people and 3 seats
How many ways can they form?
6\times5\times4
\displaystyle{P^{6}_{3}}
There are n people and r seats
How many ways can they form?
\displaystyle{P^{n}_{r}}
\displaystyle{P^{n}_{r}=\frac{n!}{(n-r)!}}
Choose r people from a group of n people
to permute
What does these two word means?
Four people are waiting in a row, how many possible ways can they form?
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
P
C
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
We have
P^{10}_3
for choosing 3 people in 10 to permute
We can get the combination by not permuting the people
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
We have
P^{10}_3
for choosing 3 people in 10 to permute
We can get the combination by not permuting the people
\frac{P^{10}_3}{3!}
We want to choose three people from a class of ten
to participate in a contest
How many ways can we choose?
We have
P^{10}_3
for choosing 3 people in 10 to permute
We can get the combination by not permuting the people
\frac{P^{10}_3}{3!}
120
We choose r people from n people
How many ways are there?
We choose r people from n people
How many ways are there?
\frac{P^n_r}{r!}
We choose r people from n people
How many ways are there?
\frac{P^n_r}{r!}
C^n_r = \frac{P^n_r}{r!} = \frac{n!}{(n-r)!r!}=\dbinom{n}{r}
Problems
Permuation
1) There are A,B,C,D,E,F six people choosing seats
A wants to sit with B
C wants to sit with D
How many ways can they sit
2) There are A,B,C,D,E,F six people choosing seats
C doesn't want to sit with D
How many ways can they sit
3) There are A,B,C,D,E,F six people choosing seats
E and F want to sit on sides
How many ways can they sit
4) We have 2, 4, 6, 8 four numbers
How many six digits number can they form?
5) We have a word COMBINATION
How many ways can you rearrange the letters
6) There are A,B,C,D,E,F six people choosing seats
C want to sit at the right side of D
How many ways can they sit
Combination
1) In a class of twenty people
Two people will be choose
How many ways can they choose
2) In a class of twenty people
We want to divide it to three groups
Each group contains 10, 6, 4 people
How many ways are there?
3) In a class of twenty people
We want to divide it to three groups
Each group contains 10, 5, 5 people
How many ways are there?
4) There are five favor of ice cream
Each person can choose two (can be same)
How many ways can one choose?
5)
6) Row three dices
How many different results
are there?
x+y+z = 10
How many positive (x, y, z)
are there?
11/13 Review
Factorial
n!
11/13 Review
Factorial
n!
The ways
n
people in a row can form
11/13 Review
Factorial
n!
The ways
n
people in a row can form
P^n_r
Permutation
11/13 Review
Factorial
n!
The ways
n
people in a row can form
P^n_r
Permutation
The ways people choose
n
r
position
11/13 Review
Factorial
n!
The ways
n
people in a row can form
P^n_r
Permutation
The ways people choose
n
r
position
Combination
C^n_r
The ways to choose
n
things from
r
What are we learning today?
What are we learning today?
Probability
Probability
What is probability?
What is probability?
The chance of an event to happen
What is probability?
The chance of an event to happen
How many times
All
P(Event)=
Example
Rowing a dice
What is the probability of showing 6
Example
Rowing a dice
What is the probability of showing 6
\frac{1}{6}
Example
Rowing two dices
What is the probability that
the sum adds up to 7
Example
Rowing two dices
What is the probability that
the sum adds up to 7
1,6
2,5
3,4
4,3
5,2
6,1
Example
Rowing two dices
What is the probability that
the sum adds up to 7
1,6
2,5
3,4
4,3
5,2
6,1
\frac{6}{6\times6} = \frac{1}{6}
Example
Rowing five dices
What is the probability that
5 will show at least once
Example
Rowing five dices
What is the probability that
5 will show at least once
(Probability of five showing at least once)
=
1 - (Probability that five do not show)
Example
Rowing five dices
What is the probability that
5 will show at least once
Answer
1-(\frac{1}{2})^5 = \frac{15}{16}
Problems
1) In an online game, when you kill a monster
You have a probability of 10% to get $10
What is the probability to get $10
if you killed 10 monsters
2) What is the probability that
a four digit number
contains a 9?
3) What is the probability that
a four digit number
contains two 9?
4) You need to write USAD Essay
in December
It will have 3 subjects from 5 for you to choose
However, you only studied 2 subjects of the 5
What is the probability that you have studied
one of the three subject
5) You and your friend are playing a game
You will choose a number from 1~10
Your friend will choose a number from 1~15
What is the probability that the number
you chosen is less than your friend
Binomial Theorem
(x+1)^1 = x+1
(x+1)^2 = x^2+2x+1
(x+1)^3 = x^3+3x^2+3x+1
(x+1)^n = ?
1
1
1
1
1
2
1
1
3
3
1
4
6
4
1
\dbinom{1}{0}
\dbinom{1}{0}
\dbinom{1}{1}
\dbinom{1}{0}
\dbinom{2}{0}
\dbinom{2}{1}
\dbinom{2}{2}
\dbinom{3}{0}
\dbinom{3}{1}
\dbinom{3}{2}
\dbinom{3}{3}
\dbinom{4}{0}
\dbinom{4}{1}
\dbinom{4}{2}
\dbinom{4}{3}
\dbinom{4}{4}
C^n_r = C^{n-1}_{r-1} + C^{n-1}_{r}
(x+1)^n = \dbinom{n}{0}x^n + \dbinom{n}{1}x^{n-1}+...
Example
x^3
What is the coefficient of
in
(2x+5)^{10}
Binomial Probability
Example
Sam is doing a test with 20 problems. However, he guessed the answer on 5 problems. He has 25% to guess the answer of the question. What is the probability that he guessed the correct answer for 3 problems
Example
Sam is doing a test with 20 problems. However, he guessed the answer on 5 problems. He has 25% to guess the answer of the question. What is the probability that he guessed the correct answer for 3 problems
C^{5}_3(0.25)^3(0.75)^2
Geometry
Vector
(Not in textbook)
What is vector?
USAD Math 2020
By sam571128
