The shape of the lesson
By the end of the lesson, learners can
- Factorials and repeats: Fill the places one at a time: \(n\) choices for the first place, \(n - 1\) for the next, and so on, giving \(n! = n(n-1)\cdots 2 \times 1\).
- Keeping things together: Glue the objects that must be together into a single block.
- Keeping things apart: For two objects that must not be together, take the total and subtract the arrangements where they ARE together.
- Fixed places and rows: Deal with the restricted places first, then fill the rest freely.
- All different: \(n!\); only \(r\) places: \({}^{n}P_{r}\).
- Repeats: divide by \(p!\) for every group of \(p\) identical objects.
- Together: glue into one unit, multiply by the arrangements inside.
- Two apart: total minus together. Several apart: gaps method.
- Overlapping restrictions: split into cases and add.
One picture every idea returns to
\(n\) different objects can be arranged in a line in \(n!\) ways. Divide by \(p!\) for each group of \(p\) identical objects. Glue objects that must be together into one unit; for objects that must be apart, subtract the 'together' count from the total or slot them into gaps.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Order matters.. How many ways can people stand in a line, or letters be arranged, when some letters repeat, some people must stay together and others must be kept apart? Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Factorials and repeats | Fill the places one at a time: \(n\) choices for the first place, \(n - 1\) for the next, and so on, giving \(n! = n(n-1)\cdots 2 \times 1\). Model the worked example Repeated letters on the board, one line at a time. Your turn: Find the number of arrangements of the letters of STATISTICS. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Keeping things together | Glue the objects that must be together into a single block. Model the worked example Two people together on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Keeping things apart | For two objects that must not be together, take the total and subtract the arrangements where they ARE together. Model the worked example Two people apart on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Fixed places and rows | Deal with the restricted places first, then fill the rest freely. Model the worked example Numbers from digits on the board, one line at a time. Your turn: How many four-digit numbers greater than 4000 and even can be made from 1, 2, 3, 4, 5, 6 with no repeats? | Numbered Heads Together |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 19 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Fixed places and rows, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Dividing by \(3! + 2!\) instead of \(3! \times 2!\). | Each repeated group contributes its own factorial to the product in the denominator. |
| Forgetting the internal arrangements of the block. | With different objects inside, multiply by their factorial; with identical objects inside (EEE), there is nothing to multiply. |
| Using 'total minus together' for three or more people who must all be apart. | Subtracting 'all three together' leaves arrangements where two of them are still next to each other. Use the gaps method instead. |
| When two restrictions overlap (first digit and last digit both even), split into cases. | A single product \(3 \times 3 \times 4 \times 3\) double-uses digits like 4 and 6. |
Permutations: arrangements in a line and with restrictions: worksheet
6 questions, 19 marks. Written in the style of the papers; no past paper question is reproduced.
Find the number of different arrangements of the letters of the word LETTER.
Find the number of arrangements of the letters of STATISTICS (a) in total, (b) which begin and end with S.
Eight people, including Ana, Ben and Cal, stand in a line. Find the number of arrangements in which Cal is at one end and Ana and Ben are not next to each other.
Four-digit numbers are made from the digits 1, 2, 3, 4, 5, 6 with no digit repeated. How many of them are even and greater than 4000?
Find the number of arrangements of the letters of BANANAS (a) in total, (b) in which no two As are next to each other.
Nine friends go to a concert and sit in two rows: 4 seats in the front row and 5 in the back row. Find the number of ways they can sit if
(a) there are no restrictions, (b) Dev and Eli must sit next to each other in the front row, (c) Dev and Eli must sit in different rows.
Answer key and marking notes
6 letters with E twice and T twice: \(\dfrac{6!}{2!\,2!} = 180\).
- (a) \(\dfrac{10!}{3!\,3!\,2!} = 50\,400\).
- (b) The middle 8 letters are S, T, T, T, A, I, I, C: \(\dfrac{8!}{3!\,2!} = 3360\).
- Cal at an end: 2 ways. The other 7 fill 7 places.
- Those 7 in total: \(7! = 5040\); with Ana and Ben together: \(6! \times 2 = 1440\).
- \(2 \times (5040 - 1440) = 7200\).
- Case 1, first digit 4 or 6: 2 choices of first digit, 2 even digits left for the last place, \(4 \times 3\) for the middle: 48.
- Case 2, first digit 5: 3 even digits for the last place, \(4 \times 3\) for the middle: 36.
- Total \(48 + 36 = 84\).
- (a) 7 letters with A three times and N twice: \(\dfrac{7!}{3!\,2!} = 420\).
- (b) Arrange B, N, N, S: \(\dfrac{4!}{2!} = 12\) ways, creating 5 gaps.
- The three As are identical, so choose 3 of the 5 gaps: \(\dfrac{5 \times 4 \times 3}{3!} = 10\).
- \(12 \times 10 = 120\).
- (a) \(9! = 362\,880\).
- (b) The front row has 3 pairs of adjacent seats, and Dev and Eli can sit in either order: \(3 \times 2 = 6\). The other 7 fill 7 seats: \(7! = 5040\). Total \(30\,240\).
- (c) Dev front and Eli back: \(4 \times 5 = 20\); or the other way round: 20. So 40 ways, then \(7!\) for the rest.
- \(40 \times 5040 = 201\,600\).
8 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the number of different arrangements of the 7 letters of the word SUCCESS.
- A420
- B5040
- C630
- D840
Eight people stand in a line. In how many ways can they stand if three particular friends must all be next to each other?
- A4320
- B720
- C2160
- D40 320
Three-digit numbers are formed from the digits 1, 2, 3, 4, 5, 6, 7, with no digit repeated. How many of these numbers are odd?
- A120
- B210
- C196
- D90
How many arrangements of the letters of PARALLEL begin and end with L?
- A360
- B60
- C720
- D3360
Five boys and three girls stand in a line. In how many ways can they stand if no two girls are next to each other?
- A14 400
- B36 000
- C2400
- D4320
Find the number of arrangements of the letters of LETTERS in which the two Ts are not next to each other.
- A900
- B540
- C360
- D1260
Four-digit numbers are formed from the digits 1, 2, 3, 4, 5, 6, 7, with no digit repeated. How many of them are even and greater than 5000?
- A160
- B180
- C120
- D40
Eight people sit in two rows of 4 seats. In how many ways can they sit if Amy and Ben must be in the same row?
- A17 280
- B8640
- C23 040
- D20 160
Quiz key
SUCCESS has S three times, C twice, and U and E once. \(\dfrac{7!}{3!\,2!} = \dfrac{5040}{12} = 420\). 5040 treats all the letters as different. The tempting 630 divides by \(3! + 2! = 8\) instead of \(3! \times 2! = 12\), and 840 allows for the repeated S but forgets the repeated C.
The block and the other 5 people make 6 units: \(6! = 720\) arrangements. The three friends can be ordered inside the block in \(3! = 6\) ways. \(720 \times 6 = 4320\). The tempting 720 forgets the arrangements inside the block, 2160 multiplies by 3 instead of \(3!\), and 40 320 is \(8!\) with no restriction.
The last digit is 1, 3, 5 or 7: 4 choices. The first two places are filled from the 6 digits left: \(6 \times 5 = 30\). Total \(4 \times 30 = 120\). 210 is \({}^{7}P_{3}\), all three-digit numbers. The tempting 196 is \(4 \times 7 \times 7\), which allows digits to repeat, and 90 uses only 3 odd digits.
PARALLEL has L three times and A twice. Putting an L at each end leaves P, A, R, A, E, L for the middle: \(\dfrac{6!}{2!} = 360\). The tempting 60 still divides by \(3!\) for the Ls, although only one L is left in the middle. 720 forgets the repeated A, and 3360 is \(\dfrac{8!}{3!\,2!}\), all arrangements with no condition.
The 5 boys can be arranged in \(5! = 120\) ways, creating 6 gaps. The three different girls go into different gaps: \(6 \times 5 \times 4 = 120\). Total \(120 \times 120 = 14\,400\). The tempting 36 000 is \(8! - 6! \times 3!\), which removes only the cases with all three girls together and still allows two girls side by side. 2400 uses \({}^{6}C_{3}\), forgetting that the girls are different, and 4320 is the number with all three girls together.
LETTERS has E twice and T twice. Total \(\dfrac{7!}{2!\,2!} = 1260\). Ts together: glue TT, leaving 6 units with two Es: \(\dfrac{6!}{2!} = 360\). Not together: \(1260 - 360 = 900\). The tempting 540 uses \(6! = 720\) for the together count, forgetting that the two Es are still identical. 360 is the together count itself, and 1260 is the total.
First digit 6: the last digit is 2 or 4 (2 choices), middle \(5 \times 4 = 20\): 40. First digit 5 or 7: the last digit is 2, 4 or 6 (3 choices), middle 20: \(2 \times 3 \times 20 = 120\). Total \(40 + 120 = 160\). The tempting 180 is the single product \(3 \times 3 \times 5 \times 4\), which lets 6 be used both first and last. 120 and 40 are the two cases on their own.
Choose their row: 2 ways. Seat Amy and Ben in 2 of its 4 seats: \(4 \times 3 = 12\) ways. The other 6 fill the 6 remaining seats: \(6! = 720\). Total \(2 \times 12 \times 720 = 17\,280\). The tempting 8640 counts only one of the two rows. 23 040 is the number with Amy and Ben in different rows, and 20 160 is \(8! \div 2\), which assumes the two cases are equally likely.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.