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K
Kate Robinson
04/12/2025
Further Maths
Permutations and Combinations
304
•
4 Dec 2025
•
Understanding Permutations and Combinations
K
Kate Robinson
@katerobinson_rafs
1 / 6
Understanding Permutations and Combinations
Think about arranging the vowels A, E, I, O, U in different orders - that's what permutations are all about! Permutations count the number of ways you can arrange elements where order matters.
When you're arranging 5 vowels, you have 5 choices for the first position, 4 for the second, 3 for the third, and so on. This gives you 5 × 4 × 3 × 2 × 1 = 120 different arrangements, which we write as 5! (5 factorial).
Combinations, on the other hand, are about selecting items where order doesn't matter. If you're choosing 3 friends from a group of 5 for a cinema trip, it doesn't matter what order you pick them in.
Quick Tip: When dealing with repeated letters (like the word DOOR), use the formula n!/(a!b!c!...) where a, b, c are the numbers of identical objects.
Partial Arrangements and Real-World Examples
Sometimes you only need to arrange part of your available options. The formula P_r = n!/! helps when you're selecting r objects from n available objects and order matters.
Let's say 6 girls and 2 boys need to line up for a photo. Without restrictions, that's 8! = 40,320 ways. But if the boys must stand together, treat them as one unit - giving you 7! arrangements, then multiply by 2! for the boys' internal arrangement.
For numbers greater than 1000 using digits 3, 4, 6, 8, 9 (no repeats), you can make 4-digit numbers plus 5-digit numbers , totalling 240 possibilities.
Combinations use the formula C_r = n!/ - perfect for selecting school council members or sports teams where order doesn't matter.
Remember: Multiplication means 'and' in probability problems, while addition means 'or'.
Pascal's Triangle and Selection Patterns
When selecting letters from ABCDE, you'll notice the number of ways follows a familiar pattern - it's actually a row from Pascal's Triangle! This connects combinations to patterns you might recognise from other maths topics.
For any selection problem, if you're choosing r objects from n available objects, you can do this in nCr ways. The formula nCr = n!/ works for everything from choosing committee members to selecting pizza toppings.
Consider a passcode with 4 characters from 26 letters and 10 digits (no repeats). The total combinations are 36C4 = 58,905. To find cases with more letters than numbers, you add the cases with 3 letters (26C3 × 10C1) plus 4 letters (26C4 × 10C0).
Your calculator probably has an nCr button - use it! These calculations can get quite large, and the calculator saves time and reduces errors.
Pro Tip: Always break complex problems into smaller cases - it makes the maths much more manageable.
From Combinations to Permutations
Here's where it gets interesting - every combination can be rearranged in multiple ways to create different permutations. If you select 3 letters like ABC, you can arrange them in 3! = 6 different orders: ABC, ACB, BAC, BCA, CAB, CBA.
Committee problems are classic combinations. Choosing 5 people from 13 candidates gives you 13C5 = 1,287 ways. For the basketball team example with democrats and republicans, remember that 'either/or' means you add the possibilities, while 'and' means you multiply.
When calculating passcode probabilities, always identify your total possible outcomes first (the denominator), then count your favourable outcomes (the numerator). The probability of more letters than numbers becomes a simple fraction once you've done the counting.
The key insight is that n distinct objects can be arranged in n! different ways. This fundamental principle underlies all permutation calculations.
Key Formula: For permutations of r objects from n distinct objects: nPr = n!/!
Handling Restrictions and Repeated Elements
Real-world arrangement problems often come with restrictions that make them trickier. When arranging Harry Potter books and Hunger Games novels with the condition that similar books stay together, treat each group as a single unit first.
For 7 Potter books and 3 Hunger Games books all staying in their groups, you have 2 groups to arrange (2! ways), then arrange within each group (7! × 3! ways). The total becomes 2! × 7! × 3! = 60,480.
Repeated elements need special handling. The word CLARA has 2 A's, so instead of 5! arrangements, you get 5!/2! = 60 arrangements. For STATISTICS with multiple repeated letters, use 10!/(3!×3!×2!).
Circular arrangements are different from linear ones - fix one person's position to avoid counting rotations as different arrangements. For 6 people in a circle, fix one person and arrange the other 5, giving 5! = 120 ways.
Strategy: For "not together" problems, calculate total arrangements minus "together" arrangements.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
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Best app on earth! no words because it’s too good
Thomas R
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Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding Permutations and Combinations
K
Kate Robinson
@katerobinson_rafs
Permutations and combinations are everywhere in your daily life - from arranging your music playlist to calculating lottery odds. Understanding these concepts will help you solve problems involving arrangements, selections, and probabilities with confidence.
Understanding Permutations and Combinations
Think about arranging the vowels A, E, I, O, U in different orders - that's what permutations are all about! Permutations count the number of ways you can arrange elements where order matters.
When you're arranging 5 vowels, you have 5 choices for the first position, 4 for the second, 3 for the third, and so on. This gives you 5 × 4 × 3 × 2 × 1 = 120 different arrangements, which we write as 5! (5 factorial).
Combinations, on the other hand, are about selecting items where order doesn't matter. If you're choosing 3 friends from a group of 5 for a cinema trip, it doesn't matter what order you pick them in.
Quick Tip: When dealing with repeated letters (like the word DOOR), use the formula n!/(a!b!c!...) where a, b, c are the numbers of identical objects.
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Partial Arrangements and Real-World Examples
Sometimes you only need to arrange part of your available options. The formula P_r = n!/! helps when you're selecting r objects from n available objects and order matters.
Let's say 6 girls and 2 boys need to line up for a photo. Without restrictions, that's 8! = 40,320 ways. But if the boys must stand together, treat them as one unit - giving you 7! arrangements, then multiply by 2! for the boys' internal arrangement.
For numbers greater than 1000 using digits 3, 4, 6, 8, 9 (no repeats), you can make 4-digit numbers plus 5-digit numbers , totalling 240 possibilities.
Combinations use the formula C_r = n!/ - perfect for selecting school council members or sports teams where order doesn't matter.
Remember: Multiplication means 'and' in probability problems, while addition means 'or'.
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Improve your grades
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Pascal's Triangle and Selection Patterns
When selecting letters from ABCDE, you'll notice the number of ways follows a familiar pattern - it's actually a row from Pascal's Triangle! This connects combinations to patterns you might recognise from other maths topics.
For any selection problem, if you're choosing r objects from n available objects, you can do this in nCr ways. The formula nCr = n!/ works for everything from choosing committee members to selecting pizza toppings.
Consider a passcode with 4 characters from 26 letters and 10 digits (no repeats). The total combinations are 36C4 = 58,905. To find cases with more letters than numbers, you add the cases with 3 letters (26C3 × 10C1) plus 4 letters (26C4 × 10C0).
Your calculator probably has an nCr button - use it! These calculations can get quite large, and the calculator saves time and reduces errors.
Pro Tip: Always break complex problems into smaller cases - it makes the maths much more manageable.
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From Combinations to Permutations
Here's where it gets interesting - every combination can be rearranged in multiple ways to create different permutations. If you select 3 letters like ABC, you can arrange them in 3! = 6 different orders: ABC, ACB, BAC, BCA, CAB, CBA.
Committee problems are classic combinations. Choosing 5 people from 13 candidates gives you 13C5 = 1,287 ways. For the basketball team example with democrats and republicans, remember that 'either/or' means you add the possibilities, while 'and' means you multiply.
When calculating passcode probabilities, always identify your total possible outcomes first (the denominator), then count your favourable outcomes (the numerator). The probability of more letters than numbers becomes a simple fraction once you've done the counting.
The key insight is that n distinct objects can be arranged in n! different ways. This fundamental principle underlies all permutation calculations.
Key Formula: For permutations of r objects from n distinct objects: nPr = n!/!
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Handling Restrictions and Repeated Elements
Real-world arrangement problems often come with restrictions that make them trickier. When arranging Harry Potter books and Hunger Games novels with the condition that similar books stay together, treat each group as a single unit first.
For 7 Potter books and 3 Hunger Games books all staying in their groups, you have 2 groups to arrange (2! ways), then arrange within each group (7! × 3! ways). The total becomes 2! × 7! × 3! = 60,480.
Repeated elements need special handling. The word CLARA has 2 A's, so instead of 5! arrangements, you get 5!/2! = 60 arrangements. For STATISTICS with multiple repeated letters, use 10!/(3!×3!×2!).
Circular arrangements are different from linear ones - fix one person's position to avoid counting rotations as different arrangements. For 6 people in a circle, fix one person and arrange the other 5, giving 5! = 120 ways.
Strategy: For "not together" problems, calculate total arrangements minus "together" arrangements.
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We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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4.8/5
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user