Combination and Permutation

Permutation means when the order is importance
The formula used is : nPr=n!/(n-r)!
When order isnt important you use combination  
The formula used is:Cr=n!/(n-r)!r!


Example:what order could 16 pool balls be in?After choosing, number "14" you can't choose it again.

Ex 2:Permutation: 16 × 15 × 14 × 13 × ... = 20,922,789,888,000

Combination:Picking a team of 3 people from a group of 10. 


C(10,3) = 10!/(7! * 3!) = 10 * 9 * 8 / (3 * 2 * 1) = 120.


Ex 3:Permutation: Picking a President, VP and Waterboy from a group of 10. 
P(10,3) = 10!/7! = 10 * 9 * 8 = 720.


Combination: Choosing 3 desserts from a menu of 10. 
C(10,3) = 120.


Permutation: Listing your 3 favorite desserts, in order, from a menu of 10.


 P(10,3) = 720.


Ex 4:Given 4 people, Bob, Mike, Sue and Alice, how many different ways can these three people be arranged where order matters?

Finding all possible arrangements of Bob, Mike, Sue and Alice where order matters would be:
BMSA,    BMAS,    BSMA,    BSAM,    BAMS,    BASM
MBSA,    MBAS,    MABS,    MASB,    MSBA,    MSAB
SBMA,    SBAM,    SMBA,    SMAB,    SABM,    SAMB
ABMS,    ABSM,    AMBS,    AMSB,    ASBM,    ASMB

There are 24 ways to arrange the four people four at a time,   or   4!

A series is a list of numbers being added. A sequence is a list of numbers. An arithmetic sequence is a sequence that is generated by adding the same number each time. Arithmetic Sequence formula tn=t1 + (n-1)d t1= first term n= term number d= difference  tn= ___ term A geometric sequence is a sequence that is generated by multiplying the same number each time. (to divide, we use fractions) formula: tn=t1(r)^(n-1) t1= first term r= what is being multiplied n= term number tn= ___ term Example: 1.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn= 4n+12 n=1 4(1)+12=16 n=2 4(2)+12=20 n=3 4(3)+12=24 n=4 4(4)+12=28 It is arithmetic because you are adding 4 each time! Example: 2.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn=-2n+8 n=1 -2(1)+8=6 n=2 -2(2)+8=4 n=3 -2(3)+8=2 n=4 -2(4)+8=0 It is arithmetic because you are adding -2 each time

Vectors in Three Dimensions!

Helloo,
I hope everyone had a wonderful weekend!

distance: square root of (x2-x1)^2  +(y2-y1)^2 + (z2-z1)^2
midpoint: ( x1+x2/2 , y1+y2/2 , z1+z2/2 )
sphere: (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2
vector equation: ( x1 y1 z1 ) = ( x0 , y0 , z0 ) + t( a , b , c )
parametric: x=x0+at  y=y0+bt  z=z0+ct
lml (magnitude): square root of x^2 + y^2 + z^2
u * v: x1 x2 + y1 y2 + z1 z2

Example 1:
Find the length and midpoint of AB.
A = ( 2 , 5 , -3 ) and B = ( 0 , 3 , 1 )
Midpoint: ( 2 + 0 / 2 , 5 + 3 / 2 , -3 + 1 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( 0 - 2 ) ^ 2 + ( 3 - 5 ) ^ 2 + ( 1 + 3 ) ^ 2 = square root of 4 + 4 + 16 = square root of 24 = 2 square root of 6

Example 2:
Find the length and midpoint of AB.
A = ( 3 , -5 , 0 ) and B = ( -1 , 1 , 2 )
Midpoint: ( 3 + 1 / 2 , -5 + 1 / 2 , 0 + 2 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( -1 - 3 ) ^ 2 + ( 1 + 5 ) ^ 2 + ( 2 + 0 ) ^ 2 = square root of 16 + 36 + 4 = square root of 56 = 2 square root of 14

Example 3:
Simplify the expression.
a. ( 3 , 8 , -2 ) + 2 ( 4 , -1 , 2 )
( 3 , 8 , -2 ) + ( 8 , -2 , 4 ) = ( 11 , 6 , 2 )
b. ( 1 , -8 , 6 ) * ( 5 , 2 , 1 )
5 + -16 + 6 = -5
c. magnitude of ( 3 , 5 , 1 )
square root of 3^2 + 5^2 + 1^2
= square root of 9 + 25 + 1
= square root of 35

Combination and Permutation

Permutation means when the order is important. You use this formula: nPr=n!/(n-r)! Combination means when the order is not important. You use this formula:Cr=n!/(n-r)!r! Example 1: 5people are standing in a line.  How many different outcomes are possible?   P(5, 5) = 5! = 5 × 4 × 3 × 2 × 1 = 120 Example 2:  In how many ways can you select a committee of 4 people from a group of 10 members? C(10, 4) = 210 Example 3: License plates for cars have to be unique. If a license plate contains 6 characters consisting of 2 letters followed by 4 digits  example:  QW2354,  A-how many different license are possible? B-how many different license are possible if letters were allowed to repeated but numbers are not allowed to be repeated? There are 26 × 26 × 10 × 9 × 8 × 7 =  3,407,040 different plates with no repeating numbers. Example 4: In how many ways can you select a committee of 3 people from a group of 12 members? C(12, 3) = 220

Function Notation

f(x) = x^2 + 5x - 2 is an example of a function. If it says "Evaluate f(2)", you can assume x = 2 and plug in 2 for all x's in the function and simplify. Here are some other things you can do with functions:

(f+g)(x) = f(x) + g(x)
(f-g)(x) = f(x) - g(x)
(f*g)(x) = f(x) * g(x)
(f/g)(x) = f(x) / g(x)
(f○g)(x) = f(g(x)) or plug in the equation g(x) into all x's in f(x)
(g○f)(x) = g(f(x)) or plug in the equation f(x) into all x's in g(x)


Ex. 1

f(x) = 3x+3
g(x) = x-8

Evaluate f(2)
f(2) = 3(2)+3
= 6+3
= 9

Evaluate g(3)
g(3) = (3)-8
= -5


Ex. 2
f(x) = 2x-8
g(x) = x^2 +4

Find (f+g)(x)
2x-8 + (x^2 + 4)
x^2 + 2x -4

Find (f-g)(x)
2x-8 - (x^2 + 4)
2x - 8 - x^2 - 4
-x^2 + 2x - 12

Find (f*g)(x)
(2x-8)(x^2+4)
2x^3+8x-8x^2-32
2x^3-8x^2+8x-32

Find (f/g)(x)
(2x-8)/(x^2+4)

Find (f○g)(x)
2(x^2+4) - 8
2x^2+8-8
2x^2

Find (g○f)(x)
(2x-8)^2 +4
4x^2-32x+64+4
4x^2-32x+68


Ex. 3
f(x) = 2x^2 -1
g(x) = x+1

Find (f○g)(2)
2(2+1)^2 -1
2(3)^2 - 1
2(9) - 1
18 - 1
17

Find (g○f)(2)
(2(2)^2 -1) +1
2(4) -1 +1
8 -1 +1
8

Find (g*f)(1)
(1+1)(2(1)^2-1)
(2)(2-1)
(2)(1)
2

Find (f+g)(8)
2(8)^2 - 1 + 8 + 1
2(64)-1+8+1
128+8
136

Find (f-g)(6)
2(6)^2 -1 -(6+1)
2(36) - 1 - 7
72 -8
64

Ch. 6

Find the equation of a circle having endpoints (3,7) and (5,-2).
center:  (3-5/2, 7-2/2)=(-1,5/2)
D:  square root of (-5-3)^2+(-2-7)^2
=square root of 64+81
=square root of 145
=12.04
=12.04/2=6.02
(x-h)^2+(y-k)^2=r^2
(x+1)^2+(y-5/2)^2=36.24

In order to find the equation for a circle, you have to find the center and the radius.  To do this, we use the midpoint formula, and the distance formula.  The midpoint formula is (x1+x2/2,y1+y2/2).  We are given tow different points, so we use those two points and plug them into to formula I just gave you.  Once you do that, you get (-1,5/2).  Now you have to use the distance formula to find the radius.  The distance formula is the square root of (x2-x1)^2+(y2-y1)^2.  You again use the two points you were given in the problem to solve for the distance.  Once you solve this, you get 12.04.  12.04 is the diameter, and we need the radius to find the equation of a circle, so all you do now is divide 12.04 by 2 and we have 6.02 now.  Now we just plug the answers we found into the circle formula.  The circle formula is (x-h)^2+(y-k)^2=r^2.  You have to square your radius you found and that will give you 36.24.  So your answer is (x+1)^2+(y-5/2)^2=36.24.

limits

**If you have a fraction with two polynomials divided, use the rules:
1. If the highest exponent of the top is equal to the highest exponent of the bottom, then you put the leading coeff over the leading coeff.

2. If the highest exponent of the top is greater than the highest exponent of the bottom, then you use plus or minus infinity (plug in to see if you get positive or negative)

3. If the highest exponent of the top is less than the highest exponent of the bottom, then it is equal to zero.

**If it doesn't follow the rules,
1. Plug into y=

2. Use the 2nd funtion, then table on your calculator.
3. Plug in 10, 100, 1000, 10000, 100000....

(In the table, if e is negative, then it is equal to zero. If e is positve, then it is equal to infinity.)

Until you see a pattern.

**If it is a geometric(a number raised to n) and |r| <1, then limit=0
If it is greater than 1, then it is infinity.

Example 1:
Find the given limit.
lim (n^2 -n)/n^3
= 0

lim [3n^2/n]
= infinity

lim (7n^2+5n)/(10n^2)
= 7/10