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A Level · Cambridge (CIE) · 9709

Select groups with exact and minimum requirements

Split a restricted selection into disjoint cases or subtract its complement, keeping group membership unordered.

Section 5.2 · Permutations and combinations

Statistics 1 · Permutations and combinations · 5.2
Exam scope and assessment notes

Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.2. State whether order matters, objects are distinguishable, repetition is permitted and positions or rows are labelled. Count mutually exclusive cases before adding them; account for internal arrangements when using blocks. These lessons include seating in two or more rows. Arrangements in a circle are excluded from this syllabus section. Unless a question says otherwise, people and named roles are distinguishable, and selection is without replacement. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

  • Calculate combinations.
  • Use product and sum rules for disjoint choices.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

For a three-person committee, which B-count cases mean at least one B?

What you will learn

  • Count selections with exact numbers from distinct groups.
  • Use cases or complements for at-least requirements.

Multiply within a specified composition

From four A-group and three B-group candidates, a three-person committee with exactly two from A and one from B has 4C2 × 3C1 = 6 × 3 = 18 possibilities. The groups are distinct and disjoint, so choosing a pair from A and one person from B specifies a unique committee.

Do not multiply by 3! unless the committee members are given distinct roles. If an individual belongs to both descriptive groups, this simple composition count may double-count or use a person twice; the stated disjoint-group assumption matters. Exactly two from A is one composition, not ‘two or more’.

At least means several disjoint possibilities

For the same seven candidates, at least one B on a three-person committee gives 4C2 × 3C1 + 4C1 × 3C2 + 4C0 × 3C3 = 18 + 12 + 1 = 31. The B-count cases 1, 2 and 3 are disjoint. Equivalently, subtract no B from all committees: 7C3 − 4C3 = 35 − 4 = 31.

A four-person team from five A and four B candidates with at least two of each must contain exactly two A and two B, because its size is four. It has 5C2 × 4C2 = 60 possibilities. Translate every condition together before listing cases; do not add impossible compositions.

Pause and explain

Four A and three B candidates: how many three-person committees have exactly two A?

Put the idea to work

Worked example

From four A and three B candidates, choose three with at least one B.

Show the worked solution
  1. All committees: 7C3 = 35.
  2. Excluded committees contain only A candidates: 4C3 = 4.
  3. Subtract 35 − 4 = 31; the complement covers every forbidden case exactly once.

Answer 31 committees.

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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

From four A and three B candidates in disjoint groups, choose three with exactly two A. Count committees.

Give me a hint

The remaining place must be filled by one B candidate.

Compare my reasoning
  1. Choose two A candidates in 4C2 = 6 ways.
  2. Choose one B candidate in 3C1 = 3 ways.
  3. Multiply to obtain 18 unordered committees.

18 committees.

Look for these in your work

  • Use the exact composition.
  • Do not order committee members.
2 · Independent

From four A and three B candidates, choose three with at least one B. Use a complement and check by cases.

Give me a hint

Subtract committees consisting only of A candidates.

Compare my reasoning
  1. 7C3 − 4C3 = 35 − 4 = 31.
  2. The disjoint cases with one, two or three B give 18, 12 and 1.
  3. Their sum is also 31, confirming the complement count.

31 committees.

Look for these in your work

  • Include all positive B counts.
  • Make the two methods agree.
3 · Transfer

A four-person team is chosen from five A and four B candidates in disjoint groups. It needs at least two of each. Count teams and explain why no other composition is possible.

Give me a hint

The two minimum counts already add to the team size.

Compare my reasoning
  1. At least two A and two B fill four places, so the composition is exactly 2 + 2.
  2. Choose the pairs in 5C2 = 10 and 4C2 = 6 ways.
  3. Multiply 10 × 6 = 60; any larger count in either group would exceed four places.

60 teams, each with two A and two B.

Look for these in your work

  • Resolve the size condition before counting.
  • Choose within each disjoint group without order.

Common mistakes

  • Keep exactly and at least distinct.
  • Use mutually exclusive group-count cases.
  • Apply all size conditions before counting.
Recall without your notes

What can you explain now?

Can a four-person team contain at least two A and at least two B in more than one composition?

Compare with the explanation

No; it must contain exactly two of each.

Their minimum counts already fill all four places.

After trying it yourself, choose your next review. This is your self-assessment.

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Make it stick

Write valid group counts, multiply within each composition, then add cases or subtract the forbidden complement.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.

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