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COUNTING OUTCOMESPERMUTATIONS & COMBINATIONS
PROBABILITYREVIEW
COUNTING OUTCOMES
Andy has 3 pairs of pants: 1 gray, 1 blue and
1 black. He has 2 shirts: 1 white and 1 red.
If Andy picks 1 pair of pants and 1 shirt, how
many different outfits does he have?
Andy can choose 1 of 3 pairs of pants and 1
of 2 shirts. A tree diagram can help you
count his choices.
8/21
The total number of choices is the product of the number of choices for item A, the number of choices for item B, etc. You can also use the counting principle.
FIRST ● SECOND ● THIRD, etc. = TOTAL NUMBER OF CHOICES
FIND THE TOTAL NUMBER OF CHOICES.
1. Moesha has 6 pairs of socks and 2 pairs of sneakers. She chooses 1 pair of socks and 1 pair of sneakers. How many possible combinations are there?
2. Kim has 5 swimsuits, 3 pairs of sandals, and 2 beach towels. In how many ways can she pick one of each to go to the beach with?
PERMUTATIONS
The expression 5! is read “5
factorial”.
It means the product of all whole
numbers from
5 to 1. 5! =
Evaluate
!3
!5
How many 3-letter codes can be made
from A, B, C, D, E, F, G, H with no
repeating letters?
This is a permutation problem.
ORDER IS IMPORTANT. ABC is
different from ACB.• There are choices for the first
letter.• There are choices for the
second letter.• There are choices for the third
letter.
PERMUTATION FORMULA
n is the number of objects and r is the number
chosen.
rnP
You can write the code as , meaning the number of permutations of 8 objects chosen at 3 times.
38P
The number of codes possible x x =
Evaluate each factorial.
Find the value of each expression.
1. !4 2. !7 3.!3
!84. !2 !5
5. 36 P 6. 25P 7. 312P 8. 415P
Solve.
9. In how many ways can you pick a football center and quarterback from 6 players who try out? 10. For a meeting
agenda, in how many ways can you schedule 3 speakers out of 10 people who would like to speak?
COMBINATIONS
Mr. Jones wants to pick 2 students from Martin, Joan, Bart, Esperanza, and Tina to demonstrate an experiment. How many different pairs of students can he choose?
In this combination problem, the ORDER DOES NOT MATTER. What are the possibilities?
There are possible combinations.
COMBINATION FORMULA
n is the number of objects and r is the number chosen
The number of combinations of 5 students taken 2 at
a time is:
25C
!r
PC rnrn
FIND THE NUMBER OF COMBINATIONS:
1. 36C 2. 49C
3. 57C 4. 34C
Solve.
5. In how many ways can Susie choose 3 of 10 books to take with her on a trip?
6. In how many ways can Rosa select 2 movies to rent out of 6 that she likes?
Probability: • Notation: P(event)
Theoretical Probability:• The likelihood of an event occurring.• Equation: # of favorable outcomes
# of total outcomes
Experimental Probability: • The number of times an event occurs in an
experiment. • Equation: # of trials an outcome occurs
total # of trials
8/19
An event who’s outcome is NOT based on a previous outcome.
DEPENDENT EVENT
An event who’s
outcome is based on
a previous outcome.
INDEPENDENT EVENT
Draw a card, keep it, then drawing another card.
Spinning a spinner, and then rolling a dice
WITH REPLACEMENT
P(A and B) = P(A) • P(B)
A bag of marbles contains 6 blue, 5 red, 3 green, 4 orange,
and 2 purple. You draw a marble at random, record your
findings, replace the marble, then draw again.
FindEx) P(blue, blue) = You Trya) P(purple, orange) = b) P(black, blue) =
100
9
10
3
10
3
WITHOUT REPLACEMENTP(A and B after A) = P(A) • P(B after A)
FindEx) P(yellow, pink) =
You Trya) P(black, black) = b) P(Nike, pink) =
95
3
19
6
10
1
A sock drawer contains 4 black, 2 yellow, 3 polk-a-dot, 5 Nike, and 6 pink. You pick a sock at random, record your findings, then pick another without replacing the first.