Counting, sample spaces and sequences
Decide whether order matters, construct the sample space and distinguish additive from multiplicative patterns.
What you will tackle
- Use sample spaces and the multiplication principle
- Distinguish permutations and combinations
- Calculate multi-stage probabilities without assuming independence
- Use arithmetic/geometric sequences and finite sums
Count arrangements and selections differently
Assigning a president and secretary from six people gives 6·5 = 30 assignments because swapping roles matters. Selecting an unordered pair gives 30/2 = 15 pairs. A tree or organised list helps make outcomes exhaustive and mutually exclusive. Without-replacement probabilities change after the first selection.
Describe the rule before extending it
An arithmetic sequence has aₙ = a₁ + (n − 1)d; a geometric sequence has aₙ = a₁r^(n − 1). The finite arithmetic sum is n(a₁ + aₙ)/2. For r ≠ 1 a geometric sum is a₁(1 − rⁿ)/(1 − r). Several observed terms can fit more than one invented rule; use the rule stated in the question.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
From six students, count assignments to president/secretary and selections of a two-person committee. A bag has 5 blue and 3 red tokens; find P(two blue without replacement).
Show a starting hint
Ask whether changing the order creates a different outcome.
Compare with the explained answer
- Distinct roles give 6·5 = 30; unordered pairs give 30/2 = 15.
- First-blue probability is 5/8, then 4/7 after one blue is removed.
- Multiply to get 20/56 = 5/14.
Answer / response criteria: 30 assignments; 15 committees; probability 5/14.
Guided 2
Find the sum of the first five terms of 2,6,18,… using both a list and the finite geometric-sum formula.
Compare with the explained answer
- The terms are 2,6,18,54,162 and sum to 242.
- a₁ = 2, r = 3 and n = 5.
- S = 2(3⁵ − 1)/(3 − 1) = 242.
Answer / response criteria: 242.
Independent practice
Independent 1
For an arithmetic sequence with first term 7 and difference 4, find term 10 and the sum of the first 10. For a geometric sequence starting 3 with ratio 2, find term 6.
Compare with the explained answer
- a₁₀ = 7 + 9·4 = 43.
- S₁₀ = 10(7 + 43)/2 = 250.
- a₆ = 3·2⁵ = 96.
Answer / response criteria: 43; 250; 96.
Independent 2
Find the sum of the first six terms of a geometric sequence starting at 3 with ratio 2.
Compare with the explained answer
- Use a₁ = 3, r = 2 and n = 6.
- S = 3(2⁶ − 1)/(2 − 1).
- This equals 3·63 = 189.
Answer / response criteria: 189.
Application practice
Application 1
An access code uses one of 4 letters followed by two different digits from 0–9. Count the codes. A security review proposes allowing repeated digits instead; find the new count and explain the change.
Compare with the explained answer
- With different digits there are 4·10·9 = 360 codes.
- Allowing repetition gives 4·10·10 = 400.
- The last position gains the option of repeating the first digit; the code remains ordered.
Answer / response criteria: 360 originally; 400 with repetition.
Application 2
A teaching model deposits $2 in week 1, then doubles the deposit each week for four weeks. Find the week-4 deposit and total deposited. Explain why the total is not the same as the last deposit.
Compare with the explained answer
- The deposits are 2,4,8,16.
- The final deposit is 16, while their sum is 30.
- A term and a cumulative sum represent different quantities.
Answer / response criteria: Week 4: $16; total: $30.
What can you explain now?
A teaching model deposits $2 in week 1, then doubles the deposit each week for four weeks. Find the week-4 deposit and total deposited. Explain why the total is not the same as the last deposit.
Compare with the explanation
Week 4: $16; total: $30.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide