## November 21, 2006

### 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

nPr = 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.