Question 1Mid
How many distinct arrangements can be made using all the letters in the word LEVEL?
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Answer: C — 30
LEVEL has 5 letters with repeated twice and repeated twice. The number of distinct arrangements is .
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Mathematics
Counting, combinations, and permutations — ACT practice questions
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What this topic is
Order decides the method in Counting, combinations, and permutations.
When order matters, as in arranging letters or awarding first and second place, use permutations; when it does not, as in picking a pair of books to read, use combinations, which divide out the repeated orderings. Words with repeated letters need extra division: BOOK has four letters, but swapping its two Os changes nothing, so half the arrangements are duplicates.
Overlapping groups call for inclusion-exclusion, adding the two group sizes and subtracting those counted in both. ACT answer choices often include both the permutation and the combination count, so deciding whether order matters comes first.
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Question 2Mid
Naomi will attend 2 of 9 workshops. How many different pairs of workshops can she choose?
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Answer: B — 36
The order of the 2 selected workshops does not matter, so use a combination. There are selections.
Question 3Easier
Of 50 students, 26 joined robotics, 18 joined drama, and 7 joined both groups. How many students joined at least one of the groups?
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Answer: C — 37
Adding the two group counts includes the 7 students in both groups twice. Inclusion-exclusion gives .
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More ACT topics
Mathematics
- Real number system and number properties
- Integer and rational exponents
- Radicals and roots
- Complex numbers
- Vectors and matrices
- Fractions, decimals, and operations
- Percents and percent change
- Ratios, rates, and proportional relationships
- Linear equations and expressions in one variable
- Inequalities (linear and compound)
- Systems of equations
- Polynomial expressions and factoring
- Quadratic equations
- Radical and rational equations
- Exponential relationships
- Function notation and evaluating functions
- Linear functions and graphs
- Quadratic and polynomial functions
- Radical and rational functions
- Piecewise functions
- Exponential and logarithmic functions
- Key features and transformations of graphs
- Lines, angles, and polygons
- Triangles — congruence, similarity, Pythagorean theorem
- Circles — arcs, sectors, angles, equations
- Surface area and volume of solids
- Coordinate geometry — distance, midpoint, slope, line equations
- Conic sections
- Right-triangle trigonometry and trig functions
- Measures of center and spread
- Data collection methods and study design
- Bivariate data and relationships (scatterplots, correlation)
- Probability and sample spaces
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