Transcript Aim: What is the counting rule?

Aim: What is the counting rule?
Exam Tomorrow
Three Rules
• Sometimes we need to know all possible
outcomes for a sequence of events
– We use three rules
1. Fundamental counting rule
2. Permutation rule
3. Combination rule
Fundamental Counting Rule
• In a sequence of n events in which the first
one has k1 possibilities and the second event
has k2 and the third has k3 and so forth, the
total number of possibilities of the sequence
will be k1 * k2 * k3 … kn
– In each case and means to multiply
Example
• A coin is tossed and a die is rolled. Find the
number of outcomes for the sequences of
events.
– Solution: Since the coin can land either heads up
or tails up and since the die can land with any one
of six numbers face up, there are 2 * 6 = 12
possibilities
• A tree diagram can also be drawn for the sequences of
events
Repetition
• When determining the number of possibilities
of a sequence of events, one must know
whether repetitions are permissible.
Example
• Example 1: The digits 0, 1, 2, 3, and 4 are to be
used in a four-digit ID card. How many different
cars are possible if repetitions are permitted?
– Solution: 5 * 5 * 5 * 5 = 54 = 625
VS
• Example 2: The digits 0, 1, 2, 3, and 4 are to be
used in a four-digit ID card. How many different
cars are possible if repetitions are not permitted?
– Solution: 5 * 4 * 3 * 2 = 120
Factorial Notation
• Uses exclamation point
– 5! = 5 *4 * 3 * 2 * 1
• For any counting n: n! = n(n-1)(n-2)(n-3)…1
• 0! = 1
Permutations
• Permutations: an arrangement of n objects in
a specific order
• The arrangement of n objects in a specific
order using r objects at a time is called a
permutation of n objects taking r objects at
atime
– It is written as nPr and the formula is
n!
n Pr 
(n  r )!
Example
• There are three choices: A, B, C of which need
to fill in 5 spots. Repetition is allowed.
– Solution:
5!
 60
5 P3 
(5  3)!
Combinations
• Combinations: a selection of distinct objects
without regard to order
• The number of combinations of r objects
selected from n objects is denoted by nCr and
is given by the formula
n!
n Cr 
(n  r )!r !
Example
• How many combinations of 4 objects are
there, taken 2 at a time?
– Solution:
4!
6
4 C2 
(4  2)!2!
Exam Tomorrow
• Complete Review Sheet
• Solutions will be on website
• Study
– In particular lessons 8 - 16