More on the fundamental principle of counting, factorial notation, and permutations.

1. It's coming ...
can you see it?
Pi by flickr user Gregory Bastien

2. Permutations of
Non-Distinguishable Objects
We're Non-Distinguishable!
I can't tell the difference!
My Lovely Twins

3. (a) How many different 4 digit numbers are there in which all the
digits are different?
(b) How many of these numbers are odd?
(c) How many of these numbers are divisable by 5?
HOMEWORK

4. (c) How many of these numbers are divisable by 5?

5. (a) How many 3-digit numbers can be formed if no digit is used more
than twice in the same number?
HOMEWORK
(b) How many of these numbers are odd?
(c) How many of these numbers are divisable by 5?

6. (c) How many of these numbers are divisable by 5?

7. In how many ways can 8 books be arranged on a shelf, if 3 particular
books must be together?
HOMEWORK

8. In how many ways can 5 people be seated in a straight line?

9. Factorial Notation
When we want to multiply all the natural numbers from a particular
number down to 1, we can use factorial notation to indicate this
operation. The symbol quot;!quot; is used to indicate factorial. This notation
can save us the trouble of writing a long list of numbers.
For example:
6! means 6 x 5 x 4 x 3 x 2 x 1 = 720
On the calculator ...
Press: [MATH]
4! = 4 x 3 x 2 x 1 = 24 [<] (Prb)
[4] (!)
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
= 3 628 800
1! = 1
By definition 0! = 1

10. In how many ways can six students be seated in 8 vacant seats?

11. Permutations (the quot;Pickquot; Formula)
A permutation is an ordered arrangement of objects.
n is the number of objects
On the calculator ...
available to be arranged
Press: [MATH]
[<] (Prb)
r is the number of objects
[2] (nPr)
that are being arranged.
Examples:
In how many ways can 5 people In how many ways can six
be seated in a straight line? students be seated in 8 vacant
seats?

12. (a) How many “words” of 4 different letters each can be made from the
letters A, E, I, O, R, S, T?
HOMEWORK
(b) How many of these words begin with a vowel and end with a
consonant?
(c) In how many of these words do vowels and consonants alternate?

13. (a) How many numbers of 5 different digits each can be formed from
HOMEWORK
the digits 0, 1, 2, 3, 4, 5, 6?
(b) How many of these numbers are even?

14. (a) In how many ways can 4 English books and 3 French books be
HOMEWORK
arranged in a row on a shelf?
(b) In how many of these ways will the French books be together?