Permutations and Combinations.notebook
Unit 2
Lesson 2
Permutations and Combinations
Permutations
A permutation is an arrangement of objects in a definite order. The number of permutations of n distinct objects is n!
Example: How many permutations are possible using the letters from the word CAT?
list:
CAT, CTA, ACT, ATC, TCA, TAC
factorials:
3! = 3 × 2 × 1 = 6
= 6
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Permutations and Combinations.notebook
Example:
1. Find the number of permutation for the letters of the word MATH.
2. Find the number of permutation for the letters of the word TRIANGLE.
Repetition
There are situations in which some of the objects that are being arranged are the same.
The number of permutations of n objects of which a objects are alike, another b objects are alike, and so on, is
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Permutations and Combinations.notebook
Example: 3. Find the number of permutations using the letters of the word STREET.
Practice
1. Find the number of permutations using the letters of the following words...
RECTANGLE
MATHEMATICS
CALCULATOR
SLEEPLESS
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Permutations and Combinations.notebook
Circular Arrangements Since circular arrangements have no distinct beginning or end, we need to adjust our permutation formula.
There are (n ­ 1)! ways to arrange n distinct objects in a circle.
Example: 4. How many ways can 5 people be arranged in a circle?
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Permutations and Combinations.notebook
Practice
2. Find the number of permutations...
a) You have your picture taken with 4 of your friends ­ side by side.
b)You go out for supper with 4 of your friends and sit at a circular table. Permutations and Grouping
In some situations not all of the objects are used in the arrangement.
The number of permutations of n objects, taking r at a time is found by
Example: How many ways could 8 runners finish 1st, 2nd, and 3rd in a race?
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Permutations and Combinations.notebook
Example
1. Rewrite each expression as a product of natural numbers. Simplify as much as possible.
5P2
9 P4
12P3
Example
2. Rewrite each expression as nPr .
7 x 6 x 5
19 x 18 x 17 x 16
57 x 56
.
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Permutations and Combinations.notebook
Practice Determine the number of permutations
3. In how many ways can a supermarket manager display 5 brands of cereals if there are 5 spaces on a shelf?
4. In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf?
Practice Determine the number of permutations
5. How many different ways can 3 red, 4 green and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets?
6. At a 100 m race, 8 runners have qualified for the final. How many ways can gold, silver and bronze be awarded?
7. Eva is playing SCRABBLE and picks the letters A, W, L, N, S, O and D. How many permutations are possible if she only uses four of her letters?
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Permutations and Combinations.notebook
8. How many ways can 6 girls and 2 boys be arranged in a row;
a) without restriction?
b) such that the 2 boys are together?
c) such that the 2 boys are not together?
Combinations
A combination is a selection of objects from a group where order is not important.
Combination of n objects taken r at a time:
Note: Combinations are a subset of permutations, thus will always have a smaller number of possibilities. Grouping items makes combinations, and rearranging within those groups makes permutations.
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Permutations and Combinations.notebook
Example
1. Rewrite each expression as a product of natural numbers. Simplify as much as possible.
5C2
6C4
29C5
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Permutations and Combinations.notebook
2. Rewrite each expression as nCr .
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Permutations and Combinations.notebook
Example 3
a)
How many 2 letter groupings (combinations) can you make from the letters of the word CAT?
b)
How many 2 letter arrangements (permutations) can you make from the letters of the word CAT?
Example 4
a) How many combinations can you make from the word TEAM picking 3 letters at a time? b) How many permutations can you make from the word TEAM picking 3 letters at a time? Comb.
Perm.
TEA
MET
MAT
MEA
TEA
MET
MAT
MEA
TAE
MTE
MTA
MAE
ATE
TEM
TAM
AME
AET
TME
TMA
AEM
EAT
EMT
AMT
EMA
ETA
ETM
ATM
EAM
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Permutations and Combinations.notebook
Practice
1. Find the number of ways in which 3 objects can be selected from a batch of 20 different objects.
2. How many different tickets are possible in the lotto game 6/49? 3. In poker, each player receives five cards. How many different hands can you make from a standard deck of 52 cards?
4. You are asked to answer 6 questions out of 10 on an exam. How many different ways can you complete the test?
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Permutations and Combinations.notebook
5. In a group of eight students, three students will be chosen to serve on a committee. a) How many different ways can this committee be formed? b) If the three students are assigned the positions of President, Vice­
President and Secretary, how many different ways can this committee be formed?
6. A university has found that there is an insufficient number of females registered in their science programs. Among a group of 35 males and 25 females applicants, the university decides to admit 10 of these individuals. Determine the number of groups of 10 people if...
a. There are no restrictions.
b. There are exactly 2 females in the group.
c. There are at least 2 females in the group.
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Permutations and Combinations.notebook
More practice...
1. How many different ways can we choose 3 pizza toppings out of a possible 7?
2. Using a standard deck of 52 cards...
a) how many 5­card hands contain all 5 cards from the same suit (a flush)?
b) how many 5­card hands contain a 4 of a kind?
3. With the 5 colors ­ red, yellow, blue, green and black, how many different ways can we color in a 3 section flag? Note: use three different colors. 4. We have create a committee of 5 people from among 10 males and 12 females. How many different committees can formed with at least 1 male. 14