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PERMUTATIONS AND COMBINATIONSHomework: Permutation and Combinations WS
WARM UP
There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green?
WARM UP- SOLUTION
There are 7 green marbles, 4 red marbles, and 2 blue marbles in the bag. Jenny picked a green marble from the bag, without replacement. What is the probability that the next marble picked is also green?
6/12 or 1/2
COMBINATIONS
Combination- order doesn’t matter. If you are dealt 5 cards from a deck it doesn’t
matter what order you get them, when you pick up your hand you have 1 combination of cards.
A combination is a grouping of the elements from a set in which the order doesn’t matter. In a combination, abc and acb would be
considered the same: The elements are the same in both groups, and the order in which they appear does not matter.
EXAMPLE 1
How many combinations are there of the letters a, b, c and d using all letters?
How many combinations are there using 3 of the letters?
EXAMPLE 1- SOLUTIONS
How many combinations are there of the letters a, b, c and d using all letters? There is 1 combination.
How many combinations are there using 3 of the letters? abc, abd, acd, bcd There are 4 combinations of 4 letters taken 3 at
a time.
EXAMPLE 2
How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time?
EXAMPLE 2- SOLUTION
How many combinations are there of the 4 letters a, b, c and d using 2 letters at a time?
ab ac ad bc bd cd
There are 6 combinations.
EXAMPLE 3
How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time?
EXAMPLE 3- SOLUTION
How many combinations are there of the 4 letters a, b, c and d using 1 letter at a time?
a b c d
There are 4 combinations.
COMBINATIONS- FORMULA
The combination of n things taken r at a time is
5! is read “five factorial”.It means (5)(4)(3)(2)(1) = 120
nCr n!
r!(n r)!
EXAMPLE 4
Find 10C6
There are lots of factors that you can cross out once you expand your factorials.
10!
6!(10 6)!10!
6!4!109876543216543214321
109874321
10371
210
EXAMPLE 5 Find 6C2 ,9C4 and 10C7.
6C2 =
9C4=
10C7=
EXAMPLE 5- SOLUTIONS Find 6C2 ,9C4 and 10C7.
6C2 =
9C4=
10C7=
6!
2!(6 2)!6!
2!4!654321214321
15
9!
4!(9 4)!9!
4!5!987654321432154321
126
10!
7!(10 7)!10!
7!3!109876543217654321321
120
EXAMPLE 6
There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer?
EXAMPLE 6-SOLUTION
There are 6 questions on Elizabeth’s essay test. She only needs to answer 2 of them, she can choose any 2 that she wants. How many different combinations of 2 test questions can Elizabeth answer?
6C2 =
6!
2!(6 2)!654321214321
15
PERMUTATIONS
A permutation is an arrangement of objects in an specific order.
Order matters. $125 is very different that $512
EXAMPLE 7
How many permutations are there using the letters ABC?
EXAMPLE 7- SOLUTIONS
How many permutations are there using the letters ABC?
ABC, ACB, BCA, CBA, BCA, BAC = 6
These are dependent events, and using the fundamental counting principle we get
3 x 2 x 1 or 3!
PERMUTATIONS- FORMULA
The permutations of n things taken r at a time is
nPr n!
(n r)!
EXAMPLE 8 Find 6P2 ,9P4 and 8P5.
6P2 =
9P4=
8P5=
EXAMPLE 8- SOLUTIONS Find 6P2 ,9P4 and 8P5.
6P2 =
9P4=
8P5=
6!
(6 2)!6!
4!6543214321
30
9!
(9 4)!9!
5!987654321
543213,024
8!
(8 5)!8!
3!87654321
321
EXAMPLE 9
Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms
a 'word', how many 'words' of any length can be formed from the letters of the word MATH?
Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.
How many ways are there to select a subcommittee of 7 members from among a committee of 17?
EXAMPLES-SOLUTIONS
Determine if each is a permutation or a combination. Assuming that any arrangement of letters forms
a 'word', how many 'words' of any length can be formed from the letters of the word MATH?
Permutation Find the number of ways to take 20 objects and
arrange them in groups of 5 at a time where order does not matter.
Combination How many ways are there to select a
subcommittee of 7 members from among a committee of 17?
Combination