Ch 2 · 5.2 Permutations and Combinations · A Level Mathematics 9709
Topic 2 of 5 in Probability & Statistics 1 — A Level Mathematics 9709 — 82 lessons.
Introduction
How many different passwords, seating plans or football teams are possible? Counting them one by one is hopeless, so we use a few counting rules. The key question every time: does the order matter? If yes, it is a permutation (an arrangement); if no, a combination (a selection). (Syllabus section 5.2, Paper 5.)
Key ideas — Arranging n different objects
n different objects can be arranged in a line in n! = n(n-1)(n-2)dots 1 ways: n choices for the first place, n - 1 for the second, and so on. Five books on a shelf: 5! = 120 ways.
🧑🏫 Teacher's explanation
Imagine you are organizing a line of friends for a photo. If you have 3 friends, you have 3 choices for who stands in the first spot. Once that person is picked, you only have 2 choices left for the middle spot, and finally 1 choice for the last spot. To find the total number of ways, you multiply these choices together: 3 × 2 × 1 = 3!.
📖 Definitions
🔍 Worked example — Easy
Find the number of ways to arrange 6 different colored flags in a line.
🔍 Worked example — Standard
How many different arrangements can be made using the letters in the word 'APPLE'?
🔍 Worked example — Exam-style
Five students, A, B, C, D, and E, are standing in a line. How many arrangements are possible if students A and B must stand next to each other?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Permutations and combinations
Choosing r objects from n different objects:
🧑🏫 Teacher's explanation
Imagine you are organizing a race. If you care about who comes in 1st, 2nd, and 3rd place, you are looking at a permutation. The order is everything; if Alice wins and Bob is second, that is a different outcome than Bob winning and Alice being second. In math, we use n! (n-factorial) to represent the total ways to arrange items, and we divide to 'remove' the arrangements we don't care about.
📖 Definitions
🔍 Worked example — Easy
How many ways can 3 students be chosen as President, Secretary, and Treasurer from a club of 8 members?
🔍 Worked example — Standard
Find the number of ways to arrange the letters in the word 'NEEDLESS'.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Repeated objects
If some objects are identical, divide by the factorial of each repeat count:
🧑🏫 Teacher's explanation
Imagine you are arranging the letters in the word 'BOB'. If the 'B's were different colors, say a red B and a blue B, you would have 3! = 6 ways to arrange them. However, because the 'B's are identical, swapping the red B with the blue B doesn't actually create a new visual arrangement. You can't tell them apart!
📖 Definitions
🔍 Worked example — Easy
Find the number of distinct arrangements of the letters in the word 'APPLE'.
🔍 Worked example — Standard
How many different ways can the letters of the word 'MISSISSIPPI' be arranged?
🔍 Worked example — Exam-style
A shelf contains 7 books: 3 identical copies of 'Maths', 2 identical copies of 'Physics', and 2 different novels. How many ways can they be arranged in a line?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Restrictions
🧑🏫 Teacher's explanation
When we arrange things, we usually assume every item is free to move anywhere. However, 'restrictions' are rules that limit this freedom. Think of it like a seating plan at a wedding: if two best friends must sit together, they act like a single unit. If they must stay apart, they are like magnets with the same poles—they need space between them.
📖 Definitions
🔍 Worked example — Easy
Find the number of ways to arrange the letters in the word 'MATH' if the letters 'M' and 'A' must be next to each other.
🔍 Worked example — Standard
Five students (A, B, C, D, E) are standing in a line. How many ways can they stand if student A and student B must NOT stand next to each other?
🔍 Worked example — Exam-style
How many ways can the letters of the word 'LEVEL' be arranged such that the word starts and ends with the letter 'L'?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Selections with conditions
For "at least" conditions, split into cases and add them (or use total - the complementary cases). Within one case, multiply the choices for each group: choosing 2 boys from 6 and 3 girls from 4 gives ^6C_2 × ^4C_3.
🧑🏫 Teacher's explanation
Imagine you are organizing a photo shoot. If you are just picking a group of friends to stand in a frame, the order doesn't matter—that is a combination. If you are deciding who stands on the left, who is in the middle, and who is on the right, the order is everything—that is a permutation.
📖 Definitions
🔍 Worked example — Easy
Find the number of ways to arrange the letters in the word 'LEVEL'.
🔍 Worked example — Standard
A committee of 5 people is to be chosen from 6 men and 4 women. How many ways can this be done if the committee must contain at least 4 women?
🔍 Worked example — Exam-style
Five students (A, B, C, D, and E) are to stand in a line for a photo. How many arrangements are possible if students A and B must NOT stand next to each other?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — People in rows
For people in two rows, treat each row as a line. Place restricted people first (e.g. choose their seats), then arrange everyone else in the remaining seats.
🧑🏫 Teacher's explanation
Imagine you are organizing a class photo. If everyone stands in one long line, you are just arranging a single sequence. But if you split the class into two rows—one sitting in front and one standing in back—you are essentially managing two separate sequences that happen at the same time.
📖 Definitions
🔍 Worked example — Easy
There are 4 people to be arranged in a single row. How many ways can they be arranged?
🔍 Worked example — Standard
6 people are to be arranged in two rows of 3. How many ways can they be arranged?
🔍 Worked example — Exam-style
5 people (A, B, C, D, E) are to be arranged in two rows: Row 1 has 3 seats and Row 2 has 2 seats. How many ways can they be arranged if A and B must be in the same row?
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: together and apart
Six friends stand in a line. In how many ways can they stand if Amir and Bina (a) must be next to each other, (b) must not be next to each other?
🔍 Worked example — Easy
Five students, A, B, C, D, and E, are standing in a line. How many ways can they be arranged if A and B must stand next to each other?
🔍 Worked example — Standard
Six friends stand in a line. In how many ways can they stand if Amir and Bina must *not* be next to each other?
🔍 Worked example — Exam-style
Seven books, including a Math book and a Physics book, are arranged on a shelf. How many arrangements are possible if the Math and Physics books must be separated by at least one other book?
⚠️ Common errors
Worked examples — Example 2: NEEDLESS with conditions
(a) The three E's together: block [EEE] plus N, D, L, S, S: 6!/2! = 360. (b) The two S's not together: S's together gives 7!/3! = 840, so 3360 - 840 = 2520.
🔍 Worked example — Easy
Find the number of distinct arrangements of the letters in the word 'LEVEL'.
🔍 Worked example — Standard
Consider the word 'NEEDLESS'. How many arrangements exist where the three E's are kept together?
🔍 Worked example — Exam-style
Consider the word 'NEEDLESS'. How many arrangements exist where the two S's are NOT together?
⚠️ Common errors
Worked examples — Example 3: a committee
A committee of 5 is chosen from 6 men and 4 women. How many committees contain at least 3 women?
🔍 Worked example — Easy
A club has 8 members. How many ways can a subcommittee of 3 be chosen?
🔍 Worked example — Standard
A committee of 5 is chosen from 6 men and 4 women. How many committees contain at least 3 women?
🔍 Worked example — Exam-style
In a group of 10 students, 6 are boys and 4 are girls. A group of 4 is chosen. How many ways can the group be formed such that it contains exactly 2 boys and 2 girls?
⚠️ Common errors
Worked examples — Example 4: two rows
Eight people sit in two rows of four. Two particular people must sit in the front row.
🔍 Worked example — Easy
How many ways can 5 people be arranged in a single line?
🔍 Worked example — Standard
Six people sit in two rows of three. Two particular people must sit in the front row. How many arrangements are possible?
🔍 Worked example — Exam-style
Eight people sit in two rows of four. Two particular people must sit in the front row. How many arrangements are possible?
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 8 letters, A×2, L×3: 8!/2! 3! = 3360. (2) block [LLL] + P, A, R, A, E: 6!/2! = 360. (3) ^7C_2 × ^5C_2 = 21 × 10 = 210. (4) 2 × 5! = 240. (5) arrange maths books 4! = 24; choose 3 of the 5 gaps in order ^5P_3 = 60: 1440.