The shape of the lesson
By the end of the lesson, learners can
- Order does not matter: Choosing Ali, Ben and Cy is the same selection as choosing Cy, Ali and Ben.
- At least, at most: cases: 'At least' and 'at most' conditions are dealt with by listing every allowed case and adding (Fig. 1).
- Particular people: If a particular person must be included, put them in and choose the rest from those left.
- Groups, then arrangements: Splitting people into named groups is a chain of selections: choose the first group, then the second from those left.
- Order irrelevant: \({}^{n}C_{r}\); order relevant: \({}^{n}P_{r} = {}^{n}C_{r} \times r!\).
- From two groups: multiply the choices from each group.
- At least / at most: list the cases and add, or total minus forbidden cases.
- Particular people: include them and choose the rest, or remove them from the pool.
- Equal unnamed groups: divide by the number of ways to swap them.
One picture every idea returns to
The number of ways of choosing \(r\) objects from \(n\) different objects, when order does not matter, is \({}^{n}C_{r} = \dbinom{n}{r} = \dfrac{n!}{r!\,(n - r)!}\). Since each selection can be ordered in \(r!\) ways, \({}^{n}P_{r} = {}^{n}C_{r} \times r!\).
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, Choose, don't order.. A committee, a team or a hand of cards is a selection: the same people in a different order is the same choice. 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: Order does not matter | Choosing Ali, Ben and Cy is the same selection as choosing Cy, Ali and Ben. Model the worked example A simple selection on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: At least, at most: cases | 'At least' and 'at most' conditions are dealt with by listing every allowed case and adding (Fig. 1). Model the worked example Adding cases on the board, one line at a time. | Pairs |
| 17 to 23 min | Teach and check: Particular people | If a particular person must be included, put them in and choose the rest from those left. Model the worked example Two who will not serve together on the board, one line at a time. Your turn: Check the first example another way: count the committees with neither Khan and those with exactly one Khan. | Pairs |
| 23 to 30 min | Teach and check: Groups, then arrangements | Splitting people into named groups is a chain of selections: choose the first group, then the second from those left. Model the worked example Groups of different sizes on the board, one line at a time. | Round Table |
| 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, 20 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 Groups, then arrangements, 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.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
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 |
|---|---|
| Using \({}^{n}P_{r}\) for a committee. | Ordered counting counts each committee \(r!\) times over: \({}^{10}P_{4} = 5040\) is 24 times too big. |
| Counting 'at least one woman' as \({}^{4}C_{1} \times {}^{8}C_{3}\): choose one woman, then any 3 of the rest. | A group with two women is then counted twice, once for each woman picked 'first'. |
| Subtracting only the committees with neither Khan (462). | The forbidden case is BOTH Khans, not neither. |
| Fixing C in the first position (\({}^{7}P_{3} = 210\)). 'Includes C' means C may be in any of the four places. | Return to Groups, then arrangements: ask a pair to state the correct version and show it on the lab before moving on. |
Combinations: selections: worksheet
6 questions, 20 marks. Written in the style of the papers; no past paper question is reproduced.
Find the values of \({}^{8}C_{3}\) and \({}^{8}P_{3}\), and explain why one is 6 times the other.
A team of 6 is chosen from 7 boys and 5 girls. Find the number of teams containing (a) exactly 2 girls, (b) at least 2 girls.
A committee of 5 is chosen from 12 people, including Mr and Mrs Lee, who refuse to serve together. Find the number of possible committees.
Ten students are divided into groups. Find the number of ways of dividing them into (a) groups of 5, 3 and 2, (b) two groups of 5.
Find the number of different selections of 4 letters that can be made from the 8 letters of the word PARALLEL.
A choir of 7 singers is chosen from 6 sopranos, 5 altos and 4 tenors. The choir must contain at least 2 singers of each type.
(a) Find the number of different choirs. (b) If instead the 7 singers were chosen at random from the 15, find the probability that the condition in (a) would be met.
Answer key and marking notes
- \({}^{8}C_{3} = 56\), \({}^{8}P_{3} = 336\).
- Each selection of 3 objects can be arranged in \(3! = 6\) orders, so there are 6 times as many ordered choices.
- (a) \({}^{5}C_{2} \times {}^{7}C_{4} = 10 \times 35 = 350\).
- (b) Total \({}^{12}C_{6} = 924\). No girls: \({}^{7}C_{6} = 7\). One girl: \({}^{5}C_{1} \times {}^{7}C_{5} = 105\).
- \(924 - 7 - 105 = 812\).
- Total \({}^{12}C_{5} = 792\); both Lees: \({}^{10}C_{3} = 120\).
- \(792 - 120 = 672\).
- (a) \({}^{10}C_{5} \times {}^{5}C_{3} = 252 \times 10 = 2520\).
- (b) \({}^{10}C_{5} = 252\) counts each split twice (the group chosen and the group left behind can be swapped), so \(252 \div 2 = 126\).
- The letters are L (3 times), A (twice), and P, R, E once each. Split by the number of Ls.
- Three Ls: the fourth letter is A, P, R or E: 4 selections.
- Two Ls: AA (1), or two different letters from A, P, R, E: \({}^{4}C_{2} = 6\). Total 7.
- One L: AA with one of P, R, E (3), or three different from A, P, R, E: \({}^{4}C_{3} = 4\). Total 7.
- No L: AA with two of P, R, E (3), or A, P, R, E (1). Total 4.
- Altogether \(4 + 7 + 7 + 4 = 22\).
- (a) Allowed cases (S, A, T): (3, 2, 2), (2, 3, 2), (2, 2, 3).
- \({}^{6}C_{3}\,{}^{5}C_{2}\,{}^{4}C_{2} = 20 \times 10 \times 6 = 1200\).
- \({}^{6}C_{2}\,{}^{5}C_{3}\,{}^{4}C_{2} = 15 \times 10 \times 6 = 900\).
- \({}^{6}C_{2}\,{}^{5}C_{2}\,{}^{4}C_{3} = 15 \times 10 \times 4 = 600\). Total 2700.
- (b) All choirs: \({}^{15}C_{7} = 6435\). Probability \(= \dfrac{2700}{6435} = 0.420\) (3 s.f.).
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.
A committee of 4 is chosen from 7 men and 5 women. How many committees contain exactly 2 women?
- A210
- B495
- C31
- D840
A club has 9 members. In how many ways can a chair, a secretary and a treasurer be chosen, if no member holds more than one post?
- A504
- B84
- C729
- D27
A group of 5 is chosen from 6 men and 4 women. How many groups contain at least one woman?
- A246
- B504
- C252
- D60
A team of 5 is chosen from 12 players, including Ravi and Sam. How many teams include Ravi but not Sam?
- A210
- B330
- C252
- D126
A team of 6 is chosen from 5 girls and 7 boys. How many teams contain at least 4 girls?
- A112
- B105
- C140
- D812
A committee of 6 is chosen from 14 people. Two of them, Mr and Mrs Patel, refuse to serve together. How many committees are possible?
- A2508
- B924
- C2079
- D495
In how many ways can 10 people be divided into a group of 4 and two groups of 3?
- A2100
- B4200
- C700
- D25 200
A code of 4 different letters is made from the letters of the word FACTORIES. How many codes contain exactly 2 vowels?
- A1440
- B60
- C240
- D3024
Quiz key
Women: \({}^{5}C_{2} = 10\). Men for the other 2 places: \({}^{7}C_{2} = 21\). \(10 \times 21 = 210\). 495 is \({}^{12}C_{4}\), every committee. The tempting 31 adds the two choices instead of multiplying, and 840 uses \({}^{5}P_{2} \times {}^{7}P_{2}\), counting each committee in every order.
The posts are distinct, so this is a permutation: \({}^{9}P_{3} = 9 \times 8 \times 7 = 504\). The tempting 84 is \({}^{9}C_{3}\), which would be right for three identical committee places. 729 is \(9^3\), allowing one person to hold several posts, and 27 is \(9 \times 3\).
Total \({}^{10}C_{5} = 252\). All men: \({}^{6}C_{5} = 6\). At least one woman: \(252 - 6 = 246\). The tempting 504 is \(4 \times {}^{9}C_{4}\): choosing one woman first and then any 4 others counts a group with two women twice. 252 is the total, and 60 is \({}^{4}C_{1} \times {}^{6}C_{4}\), exactly one woman.
Ravi fills one place and Sam is removed, so the other 4 come from the remaining 10: \({}^{10}C_{4} = 210\). The tempting 330 is \({}^{11}C_{4}\), which forgets to remove Sam. 252 is \({}^{10}C_{5}\), which forgets that Ravi has taken a place, and 126 is \({}^{9}C_{4}\), removing one person too many.
4 girls and 2 boys: \({}^{5}C_{4} \times {}^{7}C_{2} = 5 \times 21 = 105\). 5 girls and 1 boy: \({}^{5}C_{5} \times {}^{7}C_{1} = 7\). Total 112. 105 forgets the 5-girl case. The tempting 140 is \({}^{5}C_{4} \times {}^{8}C_{2}\): choosing 4 girls and then any 2 of the rest counts each 5-girl team five times. 812 is the number of teams with at most 3 girls.
Total \({}^{14}C_{6} = 3003\). Both Patels: the other 4 come from 12 people, \({}^{12}C_{4} = 495\). Allowed: \(3003 - 495 = 2508\). The tempting 924 counts only the committees with neither Patel, and 2079 subtracts those instead of the "both" committees. 495 is the forbidden case itself.
Choose the 4: \({}^{10}C_{4} = 210\). Choose 3 of the remaining 6: \({}^{6}C_{3} = 20\), and the last 3 form the other group. Swapping the two groups of 3 gives the same division, so divide by 2: \(210 \times 20 \div 2 = 2100\). The tempting 4200 forgets to divide by 2. 700 divides by \(3!\) as if all three groups could be swapped, and 25 200 multiplies by \(3!\).
Vowels A, O, I, E; consonants F, C, T, R, S. Select 2 vowels and 2 consonants: \({}^{4}C_{2} \times {}^{5}C_{2} = 6 \times 10 = 60\). Arrange the 4 chosen letters: \(4! = 24\). Total \(60 \times 24 = 1440\). The tempting 60 forgets that a code is ordered. 240 arranges the vowels among themselves and the consonants among themselves (\(2! \times 2!\)) but not the four letters together, and 3024 is \({}^{9}P_{4}\), every code.
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.