Podcast
Questions and Answers
What is the total number of permutations of the letters in the word 'KITCHEN' if it must start with a consonant and end with a vowel?
What is the total number of permutations of the letters in the word 'KITCHEN' if it must start with a consonant and end with a vowel?
How many ways can the numbers 2345678 be rearranged if the number must begin with exactly two odd digits?
How many ways can the numbers 2345678 be rearranged if the number must begin with exactly two odd digits?
What is the value of $n$ in the combination formula $nCr = \frac{n!}{r!(n-r)!}$ if $r = 3$ and $nCr = 10$?
What is the value of $n$ in the combination formula $nCr = \frac{n!}{r!(n-r)!}$ if $r = 3$ and $nCr = 10$?
How many ways can Brittany, Geoffrey, Jonathan, Kyle, Laura, and Stephanie stand in a line if boys and girls alternate with a boy starting the line?
How many ways can Brittany, Geoffrey, Jonathan, Kyle, Laura, and Stephanie stand in a line if boys and girls alternate with a boy starting the line?
Signup and view all the answers
What is the number of combinations of 5 items chosen 2 at a time?
What is the number of combinations of 5 items chosen 2 at a time?
Signup and view all the answers
How many ways can the letters in the word 'ACTIVE' be arranged if C and E must never be together?
How many ways can the letters in the word 'ACTIVE' be arranged if C and E must never be together?
Signup and view all the answers
How many ways can you arrange 5 different books on a shelf if the first and last books are already chosen?
How many ways can you arrange 5 different books on a shelf if the first and last books are already chosen?
Signup and view all the answers
How many committees of 3 can be formed from 10 people if one person refuses to work with another person?
How many committees of 3 can be formed from 10 people if one person refuses to work with another person?
Signup and view all the answers
If a password consists of 3 uppercase letters and 4 digits, how many different passwords are possible if repetition is allowed?
If a password consists of 3 uppercase letters and 4 digits, how many different passwords are possible if repetition is allowed?
Signup and view all the answers
How many ways can you arrange 5 people in a row if 2 people must always be next to each other?
How many ways can you arrange 5 people in a row if 2 people must always be next to each other?
Signup and view all the answers
How many different ways can you select 4 books from a shelf of 10 books if the first and last books are already chosen?
How many different ways can you select 4 books from a shelf of 10 books if the first and last books are already chosen?
Signup and view all the answers
If a license plate consists of 3 uppercase letters and 2 digits, how many different license plates are possible if repetition is not allowed?
If a license plate consists of 3 uppercase letters and 2 digits, how many different license plates are possible if repetition is not allowed?
Signup and view all the answers
If a committee consists of three members, with two members selected from a group of five men and one member selected from a group of four women, how many different committees can be formed?
If a committee consists of three members, with two members selected from a group of five men and one member selected from a group of four women, how many different committees can be formed?
Signup and view all the answers
If a box contains 12 identical pens, and 8 identical pencils, how many ways are there to choose 4 items from the box?
If a box contains 12 identical pens, and 8 identical pencils, how many ways are there to choose 4 items from the box?
Signup and view all the answers
If a student can choose to play either tennis or basketball, and there are 3 tennis courts and 2 basketball courts, how many choices does the student have?
If a student can choose to play either tennis or basketball, and there are 3 tennis courts and 2 basketball courts, how many choices does the student have?
Signup and view all the answers
If there are 5 events, A, B, C, D, and E, and events A and B are disjoint, and events C and D are disjoint, how many ways are there for at least one of the events to occur?
If there are 5 events, A, B, C, D, and E, and events A and B are disjoint, and events C and D are disjoint, how many ways are there for at least one of the events to occur?
Signup and view all the answers
If a password consists of 2 uppercase letters, and there are 26 uppercase letters, how many possible passwords are there?
If a password consists of 2 uppercase letters, and there are 26 uppercase letters, how many possible passwords are there?
Signup and view all the answers
If a committee consists of 3 people chosen from a group of 10 people, how many ways are there to choose the committee?
If a committee consists of 3 people chosen from a group of 10 people, how many ways are there to choose the committee?
Signup and view all the answers
Study Notes
Conditional Permutations
- To find the number of permutations with conditions, identify the restrictions and calculate the possible arrangements.
- Example: In the word "ACTIVE", if C & E must always be together, there are 23 possible arrangements.
- Example: In the word "ACTIVE", if C & E must never be together, there are 24 possible arrangements.
Combinations
- A combination is an arrangement of items where order does not matter.
- The formula to find the number of combinations of items chosen at a time is 𝑛! / (𝑟!(𝑛-𝑟)!) where 0≤𝑟 ≤𝑛.
- Example: How many 4-letter words can be created if repetitions are not allowed?
- Example: How many three-letter words can be made from the letters of the word KEYBOARD?
Combinations Examples
- If there are 35 songs and you want to make a mix CD with 17 songs, there are many ways to arrange them.
- If there are six different colored balls in a box, and you pull them out one at a time, there are many ways to pull out 4 balls.
- A committee is to be formed with a president, a vice-president, and a treasurer from 20 people, and there are many different committees possible.
Permutations with Conditions
- Permutations with specific positions require analyzing how many possible ways each space can be filled.
- Example: How many numbers can be made from rearranging 2345678 if the number must begin with two odd digits?
- Permutations with always together conditions require treating the objects together as 1 and determining the number of arrangements, and then finding the number of internal arrangements.
Lesson Outcomes
- Apply the "Sum Rule" Principle and "Product Rule" Principle.
- Distinguish between "Permutations and Combinations".
- Calculate Permutations and Combinations.
Combinatorial Analysis
- Combinatorial Analysis includes the study of permutations and combinations.
- It is concerned with determining the number of logical possibilities of some event without necessarily identifying every case.
Sum Rule Principle
- If some event can occur in 𝑚 ways and a second event can occur in 𝑛 ways, and suppose both events cannot occur simultaneously, then it can occur in 𝑚 + 𝑛 ways.
- Example: Computing faculty must choose either a student or a faculty member as a representative for a university committee, and there are 27 faculty members and 83 CS majors.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of combinatorics and permutations with these questions on arranging words, songs, and committee members. Practice your problem-solving skills with these challenging scenarios!
