Ok, so my math teacher didn't tell us ways to do this faster, and it's late enough as it is so I don't want to be up all night doing homework. D;
*Note: A permutation is where the order matters. A combination is where it doesn't.
Find the number of permutations.
1.Ways to arrange the letters in the word GYMNAST
2. Ways to arrange 8 books on a bookshelf
3.Ways to choose your first, second, and third favorite movies from a list of 30 movies
Find the number of combinations.
4. Ways to choose 4 different flowers from the following kinds: rose, tulip, daisy, petunia, sunflower, begonia, lilac, orchid
5. Ways to choose 6 different baseball cards from 25 baseball cards
6. Ways to choose 5 different magazines from 34 magazines
In Exercises 7-10, tell whether the situation describes a permutation or a combination. Then answer the question.
7. A teacher is grouping people together to work on a project. There are 36 people in the class and each group will have 3 people. How many ways can the teacher choose the first group?
8. You are burning a CD of your 10 favorite songs. How many ways can you order the songs on the CD?
9.You are at the video store to rent video games. The store has 18 games that you like, but you can only rent 5 games. How many ways can you rent the 5 games?
10. At a basketball tournament, you want to assign each team with a different number so that you can easily come up with a game schedule. If there are 8 teams in the tournament, how many different ways can you number the teams?
11. In how many different ways can the letters in the word OBOE be arranged? Explain your reasoning.
12. Your classes are being scheduled for the upcoming school year. You will be taking English, math, history, physical education, Spanish, woodshop, and computer basics. What is the probability that woodshop will be scheduled during first period and math will be scheduled during second period?
1. Ways to arrange the letters in the word GYMNAST: There are 7 letters in the word GYMNAST. So, the number of permutations is 7!
2. Ways to arrange 8 books on a bookshelf: There are 8 books, so the number of permutations is 8!
3. Ways to choose your first, second, and third favorite movies from a list of 30 movies: Since the order matters, this is a permutation problem. The number of permutations is 30 P 3.
4. Ways to choose 4 different flowers from the given kinds: Since the order does not matter, this is a combination problem. The number of combinations is 8 C 4.
5. Ways to choose 6 different baseball cards from 25 baseball cards: This is also a combination problem. The number of combinations is 25 C 6.
6. Ways to choose 5 different magazines from 34 magazines: Again, this is a combination problem. The number of combinations is 34 C 5.
7. A teacher is grouping people together to work on a project: Since the order of the people in the group does not matter, this is a combination problem. The number of combinations is 36 C 3.
8. You are burning a CD of your 10 favorite songs: Since the order of the songs on the CD matters, this is a permutation problem. The number of permutations is 10!.
9. You are at the video store to rent video games: Again, this is a combination problem. The number of combinations is 18 C 5.
10. At a basketball tournament: Since the teams need to have different numbers, this is a permutation problem. The number of permutations is 8!.
11. In how many different ways can the letters in the word OBOE be arranged? The word OBOE has 4 letters. Since there are repetitions (the letter O appears twice), we need to adjust the formula. The number of permutations is 4! / (2! x 1!) = 12.
12. Your classes are being scheduled for the upcoming school year: The probability of woodshop being scheduled during first period and math being scheduled during second period depends on the total number of periods and the number of possible schedules. Can you provide more information?
To find the number of permutations and combinations, we can use the formulas:
Permutations (nPr) = n! / (n - r)!
Combinations (nCr) = n! / (r!(n - r)!)
Now, let's solve each problem step-by-step:
1. Ways to arrange the letters in the word GYMNAST - Since it is a word with no repeated letters, we can use the formula for permutations. There are 7 letters, so n = 7. We need to arrange all 7 letters, so r = 7. Therefore, the number of permutations is 7P7 = 7!
2. Ways to arrange 8 books on a bookshelf - Again, we can use the formula for permutations. There are 8 books, so n = 8. We need to arrange all 8 books, so r = 8. Therefore, the number of permutations is 8P8 = 8!
3. Ways to choose your first, second, and third favorite movies from a list of 30 movies - Since the order matters, we need to use permutations. There are 30 movies to choose from, so n = 30. We need to choose 3 movies, so r = 3. Therefore, the number of permutations is 30P3 = 30! / (30 - 3)!
4. Ways to choose 4 different flowers from the following kinds: rose, tulip, daisy, petunia, sunflower, begonia, lilac, orchid - Since the order doesn't matter, we need to use combinations. There are 8 flowers to choose from, so n = 8. We need to choose 4 flowers, so r = 4. Therefore, the number of combinations is 8C4 = 8! / (4!(8 - 4)!)
5. Ways to choose 6 different baseball cards from 25 baseball cards - Again, we need to use combinations since the order doesn't matter. There are 25 cards to choose from, so n = 25. We need to choose 6 cards, so r = 6. Therefore, the number of combinations is 25C6 = 25! / (6!(25 - 6)!)
6. Ways to choose 5 different magazines from 34 magazines - The order doesn't matter, so we use combinations. There are 34 magazines to choose from, so n = 34. We need to choose 5 magazines, so r = 5. Therefore, the number of combinations is 34C5 = 34! / (5!(34 - 5)!)
Let's move on to the next set of problems:
7. A teacher is grouping people together to work on a project. There are 36 people in the class and each group will have 3 people. Since the order of the group doesn't matter, we use combinations. There are 36 people to choose from, so n = 36. We need to choose 3 people for the group, so r = 3. Therefore, the number of combinations is 36C3 = 36! / (3!(36 - 3)!).
8. You are burning a CD of your 10 favorite songs. Since the order matters, we need to use permutations. There are 10 songs to choose from, so n = 10. We need to arrange all 10 songs, so r = 10. Therefore, the number of permutations is 10P10 = 10!.
9. You are at the video store to rent video games. The store has 18 games that you like, but you can only rent 5 games. Since the order doesn't matter, we use combinations. There are 18 games to choose from, so n = 18. We need to choose 5 games, so r = 5. Therefore, the number of combinations is 18C5 = 18! / (5!(18 - 5)!).
10. At a basketball tournament, you want to assign each team with a different number so that you can easily come up with a game schedule. If there are 8 teams in the tournament, we need to use permutations since the order matters. There are 8 teams to assign numbers to, so n = 8. We need to assign numbers to all 8 teams, so r = 8. Therefore, the number of permutations is 8P8 = 8!.
11. In how many different ways can the letters in the word OBOE be arranged? Since the word contains repeated letters, we need to adjust the formula for permutations. The word "OBOE" has a length of 4, but it contains two identical letters 'O'. So, the number of different arrangements is 4! / 2!.
12. To find the probability, we need to determine the total number of possible schedules and then calculate the probability of the desired schedule occurring. Assuming each period has only one available time slot for each subject, the total number of possible schedules is the product of the number of periods for each subject. So, the total number of possible schedules is 7! (number of periods for each subject).
To calculate the probability of the desired schedule, we consider that there is only one possible order for the desired schedule of woodshop during the first period and math during the second period. Therefore, the probability is 1 / 7!.
I hope this step-by-step explanation helps you understand how to find the number of permutations and combinations, and how to identify whether a situation describes a permutation or a combination. If you have any further questions, feel free to ask!
To find the number of permutations, you can use the formula:
n! / (n-k)!
where n is the total number of items and k is the number of items you are arranging or choosing.
1. To find the number of ways to arrange the letters in the word GYMNAST, we have 7 letters. So n = 7. There are no specific items we are choosing, so k = 7. Using the formula, the number of permutations is 7! / (7-7)! = 7!.
2. To find the number of ways to arrange 8 books on a bookshelf, we have 8 books. So n = 8. We are arranging all the books, so k = 8. Using the formula, the number of permutations is 8! / (8-8)! = 8!.
3. To find the number of ways to choose your first, second, and third favorite movies from a list of 30 movies, we have 30 movies. So n = 30. We are choosing 3 movies, so k = 3. Using the formula, the number of permutations is 30! / (30-3)! = 30! / 27!.
To find the number of combinations, you can use the formula:
n! / (k! * (n-k)!)
4. To find the number of ways to choose 4 different flowers from the given kinds, we have 8 kinds of flowers to choose from. So n = 8. We are choosing 4 flowers, so k = 4. Using the formula, the number of combinations is 8! / (4! * (8-4)!) = 8! / (4! * 4!).
5. To find the number of ways to choose 6 different baseball cards from 25 baseball cards, we have 25 cards to choose from. So n = 25. We are choosing 6 cards, so k = 6. Using the formula, the number of combinations is 25! / (6! * (25-6)!) = 25! / (6! * 19!).
6. To find the number of ways to choose 5 different magazines from 34 magazines, we have 34 magazines to choose from. So n = 34. We are choosing 5 magazines, so k = 5. Using the formula, the number of combinations is 34! / (5! * (34-5)!) = 34! / (5! * 29!).
Now let's determine whether each situation is a permutation or a combination and answer the questions:
7. A teacher is grouping people together to work on a project. Each group will have 3 people. This situation describes choosing groups, so it is a combination. The question asks how many ways the teacher can choose the first group, which means they are arranging the groups. So it is a permutation. To find the number of ways to arrange the groups, we have 36 people to choose from, so n = 36. We are choosing groups of 3, so k = 3. Using the formula, the number of permutations is 36! / (3! * (36-3)!) = 36! / (3! * 33!).
8. You are burning a CD of your 10 favorite songs. This situation describes choosing the order of songs, so it is a permutation. The question asks how many ways you can order the songs on the CD, so it is a permutation. To find the number of ways to arrange the songs, we have 10 songs, so n = 10. We are arranging all the songs, so k = 10. Using the formula, the number of permutations is 10! / (10-10)! = 10!.
9. You are at the video store to rent video games. You can only rent 5 games. This situation describes choosing games, so it is a combination. The question asks how many ways you can rent the 5 games, so it is a combination. To find the number of ways to choose the games, we have 18 games to choose from, so n = 18. We are choosing 5 games, so k = 5. Using the formula, the number of combinations is 18! / (5! * (18-5)!) = 18! / (5! * 13!).
10. At a basketball tournament, you want to assign each team with a different number. This situation describes choosing numbers, so it is a combination. The question asks how many different ways you can number the teams, so it is a combination. To find the number of ways to choose the numbers, we have 8 teams to choose from, so n = 8. We are choosing numbers for all the teams, so k = 8. Using the formula, the number of combinations is 8! / (8! * (8-8)!) = 8! / (8! * 0!).
11. To find the number of different ways the letters in the word OBOE can be arranged, we can use the formula for permutations. The word OBOE has 4 letters, so n = 4. We are arranging all the letters, so k = 4. Using the formula, the number of permutations is 4! / (4-4)! = 4!.
12. To find the probability that woodshop will be scheduled during first period and math will be scheduled during second period, we need to know the total number of possible schedules. In this case, the order of the classes matters, so it is a permutation. The total number of possible schedules is the total number of permutations of all the classes: 7! / (7-7)! = 7!. To find the probability of woodshop being scheduled during first period and math during second period, we need to divide the number of ways this specific combination can occur by the total number of possible schedules. The specific combination of woodshop and math being scheduled in those periods is only one way, so the probability is 1 / 7!.
My friend if you don't care about your own math work, then no one else will, too.
Math rules!