9.4 Counting Methods, Combinations, Permutations, Venn & Normal Distributions

Counting Methods, Combinations, Permutations, Venn & Normal Distributions

Quick Answer: Four tools handle every GRE counting/set/distribution question: the multiplication rule (n_1 × n_2 × ... × n_k), permutations P(n, k) = n!/(n − k)! when order matters, combinations C(n, k) = n!/(k!(n − k)!) when order does not matter, and inclusion-exclusion via Venn diagrams. The normal distribution is the only continuous model tested — symmetric, mean = median = mode, with the 68–95–99.7 empirical rule and z = (x − μ)/σ for standardization. Inferential statistics is out of scope.

The Multiplication Rule of Counting

If a task can be broken into k stages, with n_i choices at stage i, the total number of outcomes is:

Total = n_1 × n_2 × ... × n_k

This is the foundation of all counting on the GRE.

Worked Example — Meal Combinations

A restaurant offers 4 appetizers, 5 main courses, and 3 desserts. How many three-course meals are possible? 4 × 5 × 3 = 60.

Worked Example — License Plates

A license plate has 3 letters followed by 4 digits (repetition allowed). Number of plates = 26 × 26 × 26 × 10 × 10 × 10 × 10 = 26³ × 10⁴ = 17,576 × 10,000 = 175,760,000.

Factorials

n! = n × (n − 1) × (n − 2) × ... × 2 × 1, with 0! = 1 by convention.

Factorials count arrangements of n distinct objects in a line. 5! = 120, so five distinct books can be arranged on a shelf in 120 ways.

Permutations — Order Matters

A permutation is an ordered selection of k objects from n distinct objects:

P(n, k) = n! / (n − k)!

Worked Example — Race Rankings

Eight runners compete; gold, silver, and bronze are awarded (no ties). How many orderings of the top 3?

• P(8, 3) = 8! / 5! = 8 × 7 × 6 = 336.

The intuition: 8 choices for gold, then 7 for silver, then 6 for bronze → 8 × 7 × 6 = 336.

Combinations — Order Does Not Matter

A combination is an unordered selection of k objects from n distinct objects:

C(n, k) = n! / (k! × (n − k)!)

The extra k! in the denominator removes the orderings counted in a permutation.

Worked Example — Committee Selection

From 10 people, choose a committee of 4. Order does not matter.

• C(10, 4) = 10! / (4! × 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 5,040 / 24 = 210.

Choosing Permutation vs. Combination

Ask: "Would selecting the same items in a different order count as a different outcome?" If yes → permutation. If no → combination.

Venn Diagrams and Inclusion-Exclusion

A Venn diagram represents overlapping sets as circles inside a universal set. For two sets:

|A ∪ B| = |A| + |B| − |A ∩ B|

The subtraction removes the double-counted overlap.

Worked Example — Two-Set Survey

In a survey of 100 people: 60 like coffee, 40 like tea, and 20 like both. How many like neither?

• |A ∪ B| = 60 + 40 − 20 = 80.
• Neither = 100 − 80 = 20.

Three-Set Inclusion-Exclusion

For three sets A, B, C:

|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|

The alternating signs correct for over- and under-counting at each level of overlap.

Worked Example — Three-Set Venn

In a class of 50 students: 25 take math, 20 take physics, 18 take chemistry; 8 take math and physics, 6 take math and chemistry, 5 take physics and chemistry; 2 take all three. How many take at least one?

• |A ∪ B ∪ C| = 25 + 20 + 18 − 8 − 6 − 5 + 2 = 46.
• None of the three = 50 − 46 = 4.

The Normal Distribution

The normal distribution is a continuous, symmetric, bell-shaped probability distribution characterized by its mean μ and standard deviation σ. Key properties:

• Symmetric about the mean; mean = median = mode.
• Total area under the curve = 1 (it is a probability distribution).
• Tails approach the x-axis but never touch it (asymptotic).

The Empirical (68–95–99.7) Rule

For a normal distribution:

This means about 34% lies between μ and μ + 1σ, about 13.5% between μ + 1σ and μ + 2σ, and about 2.5% above μ + 2σ.

Worked Example — Empirical Rule

Test scores are normally distributed with mean 70 and SD 5. What percent of scores fall between 60 and 80?

• 60 = μ − 2σ and 80 = μ + 2σ.
• By the empirical rule, about 95% of scores fall within 2σ of the mean, so about 95% lie between 60 and 80.

Z-Scores — Standardization

A z-score measures how many standard deviations a value x is from the mean:

z = (x − μ) / σ

Positive z: above mean. Negative z: below mean. z = 0: equal to mean.

Worked Example — Comparing Across Distributions

SAT scores: mean 1050, SD 100. ACT scores: mean 21, SD 5. Alice scores 1250 on the SAT; Bob scores 31 on the ACT. Who performed better relative to their group?

• Alice's z = (1250 − 1050)/100 = 2.0 (two SDs above mean).
• Bob's z = (31 − 21)/5 = 2.0 (two SDs above mean).
• Same z-score → equally strong relative performance.

What the GRE Does NOT Test

The Quant section covers descriptive statistics and basic probability — NOT inferential statistics. You will not see:

• Hypothesis testing (null/alternative, p-values, Type I/II errors).
• Confidence intervals.
• Regression inference (slope significance, R² interpretation beyond "closer to 1 fits better").
• t-distributions, chi-square tests, ANOVA.
• Sampling distributions of the sample mean.

If a question describes a sample and asks about a population parameter, read carefully — the GRE may ask only about the sample itself (descriptive) or about basic probability, never about inference.

Strategy Summary

• Counting: multiplication rule first, then permutation (order) or combination (no order).
• Sets: Venn diagram, inclusion-exclusion with alternating signs.
• Normal: identify μ and σ, apply the empirical rule, or standardize with z.
• Probability distributions: only the normal is in scope; do not invoke binomial formulas unless the question explicitly defines one (rare).