### scribe post: counting

Hello guys... This is **Richard** and it's my third scribe post. But before we proceed to our lesson for today, let's take a look to the notes on last thursday (November 16) since there's no scribe that day.

**COUNTING**(a.k.a.

**COMBINATORICS**)

*The Fundamental Principle of Counting*

If there are

**m**ways to do the first thing and**n**ways to do a second thing then there are**m****•****n**ways to do both things.example:

How many different outfits can be made if a person has 3 pairs of pants and 4 shirts?

solution:

3 ways to choose pants

4 ways to choose shirts

۞ 3 x 4 = 12 ways to choose both

*Factorial Notation*

Definition:

**n! = n (n-1) (n-2) (n-3) . . . 3, 2, 1**

**0! = 1**

examples:

**4!**= 4 • 3 • 2 • 1

= 24

**6!**= 6 • 5 • 4 • 3 • 2 • 1

**=**720

**Note: FACTORIAL NOTATION IS UNDEFINED FOR NEGATIVE NUMBERS**

n! = n (n-1)!

*Permutation*

An order arrangement of objects.

FORMULA

n

**P**r = n!/ (n-r)!n

**is the number of objects to pick from****r**is the number of object to be arrange

nPr is read as " n Pick r"

nPr given a set of

**n**objects. How many orderd arrangements can be made using only**r**of them at a time.examples:

There are 8 horses in a race. In how many different ways can they finish 1st, 2nd, 3rd.

solution:

(by fundamental principle of counting)

8 • 7 • 6

- - - = 336

1 2 3

(by Permutation)

8P3 = 8! / (8-3)!

= 8!/ 5!

= 336

*Permutation of Non-distinguishable objects*

The number of ways to arrange "n" objects that contain K1, K2, K3. . . .etc. Set of non-distinguishable objects is given by:

**n!**

**__________**

**K1! K2! K3!...**

example:

How many different words can be made from the letters in the words:

(a) book (b) Mississippi

(a) # letters = 4

# of O's = 2

= 4! / 2!

= 12

(b) # letters = 11

# of I's = 4

# of S's = 4

# of P's = 2

= 11! / (2! 4! 4!)

November 21, 2006

In the beginning of the class, Mr.K give us the following problems without using a calculator.

Simplify

(a) 8! / 4!

8•7•6•5•

*4!*_______ 4! is reduce

*4!*

so we left with 8 • 7 • 6 • 5

*=*1680

(b) 7! - 5!

7 • 6 • 5! - 5!

5! (7 • 6 - 1) factoring by 5!

120 (41) = 4920

(c) 10! 4! / 8! 6!

10•9•8! 4!

_______ 8! & 4! are reduce

8! 6•5•4!

10•9

____ = 3

6•5

Write using factorial

(a) 5•4•3•2•1

= 5!

(b) 5•4•3

= 5! / 2!

(c) 20•19•18•17

=20! / 16!

Simplify

(a) (n+1)! / n!

(n+1) n!

______ from the definition of factorial notation n! = n ( n-1)!

n!

= n + 1

(b) (n-1)! / n!

(n-1)!

_____

n (n-1)!

= 1 / n

(c) (n+1) (n-1) / (n!)²

(n+1) n! (n-1)!

__________

n! n (n-1)!

= (n+1) / n

(d)

(2n-1)! (n+1)!

__________

(2n+1)! (n-1)!

(2n-1)! (n+1)( n ) (n-1)!

_________________

(2n+)(2n)(2n-1)! (n-1)!

n(n+1)

_______ n is reduce

2n(2n+1)

n+1

____

4n+2

SOLVE:

(a) 3! (n-1)!

_______ = 72

(n-3)!

(n-1) ! 72

____ = __

(n-3)! 6

(n-1)(n-2) = 12 (n-3)! is reduce.

n² -3n +2 = 12

n²-3n-10 = 0

(n-5) (n+2) = 0

n= 5 n= -2

-2 is rejected because the factorial of negative numbers is undefined.

Always indicate if it is accepted or reject.

(b) (2n-1)!

________ = 10

2! (2n -3)

(2n-1) (2n-2) (2n-3)

______________ = 20

(2n-3)

(2n-1) (2n-2) = 20

4n² -6n - 18 = 0

2n² -3n - 9 = 0 it is divided by 2

(2n+3) (n-3)

n= -3/2 n = 3

-3/2 is rejected

I'm sorry guys if this is not nice..I'll fix it later. I just can't upload the images.

By the way the next scribe will be John.

what an awesome post. I like the boldings and colours. Great job, this gets my vote for hall of fame.

ReplyDeleteawesome scribe post is uber cool. you got my vote for the scribe post hall of fame

ReplyDeleteit was a great scribe.. the colors are helpful for me to understand the lesson.. that's why you got my vote.. keep it up..

ReplyDeleteawesome job ricardo! i like the spacings and colors you used. sorry i didn't comment earlier but gets my vote for hall of fame. ^_^

ReplyDelete