Counting cases and simple probability · GMAT Quant

Counting starts by deciding whether order matters and whether categories overlap. A probability compares favorable cases with the relevant total of equally likely cases.

A reliable method

Use combinations for unordered teams, arrangements for distinct positions, and inclusion-exclusion for overlapping groups. Count a small example to confirm the interpretation.

The common trap

Selecting two people is different from assigning two roles. Adding group sizes double-counts members in both groups unless the overlap is subtracted.

Check the conclusion

Check that every outcome is counted once. Keep probabilities between zero and one, and state the random-selection assumption rather than assuming all real-world outcomes are equally likely.

Contrast the cases

Choosing two employees from six gives 15 unordered teams, while assigning president and secretary to different employees gives 30 ordered choices. When two specific books among four must be adjacent, treat them as one block: three objects have 3! arrangements, multiplied by two internal block orders, giving 12. Counting only the block misses this orientation factor. An overlap problem instead needs inclusion-exclusion, not a permutation formula.

Transfer and explain

Transfer: among twenty employees, ten attend A, eight attend B and three attend both. The union has 10+8−3=15, leaving five attending neither. Explain why subtracting both attendance counts without adding back overlap gives an incorrect result.

WORKED EXAMPLE

Of 36 employees, 18 attend workshop A, 15 attend B, and 6 attend both. How many attend neither?

Count at least one workshop by subtracting the overlap once.
The union is 18+15−6 = 27.
Subtract from 36 to obtain 9 attending neither.

Answer: 9

Compare the other answer choices

Returns the overlap.

Removes both-only membership from the union as well.

Counts everyone outside A, including B-only employees.

Returns the union rather than neither.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

GMAT exam content Siyavula Grade 10 mathematics foundation reference