1. Permutations & Combinations Fundamental Formulas

Factorial: n! = n × (n - 1) × ... × 1, with 0! = 1. Permutation (Arrangement where order matters): ⁿPᵣ = n! / (n - r)!. Combination (Selection where order does not matter): ⁿCᵣ = n! / [r!(n - r)!]. Important relations: ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's Identity); ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ. Number of permutations of n objects with p identical of one kind and q identical of another = n! / (p! · q!).

2. Classical Probability & Addition/Multiplication Theorems

Probability of event E: P(E) = n(E) / n(S) (favorable outcomes / total outcomes in sample space S). Range: 0 ≤ P(E) ≤ 1. Complementary event: P(E') = 1 - P(E). Addition Theorem for any two events A and B: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually exclusive events (where A ∩ B = ∅): P(A ∪ B) = P(A) + P(B). Independent events: P(A ∩ B) = P(A) · P(B).

3. Conditional Probability & Bayes' Theorem

Conditional probability of event A given that B has occurred: P(A|B) = P(A ∩ B) / P(B) (provided P(B) > 0). Bayes' Theorem for partition events E1, E2, ... En and given observed evidence A: P(Ei|A) = [P(Ei) · P(A|Ei)] / [Σ P(Ej) · P(A|Ej)]. Allows calculating posterior probability by updating prior probability with conditional test evidence.

4. Measures of Central Tendency: Mean, Median & Mode

Arithmetic Mean: For ungrouped data: x̄ = (Σ xi) / N. For grouped frequency data: x̄ = (Σ fi·xi) / (Σ fi). Median: Middle value of sorted data: For odd N: Term ((N + 1)/2). For even N: Average of (N/2)th and (N/2 + 1)th terms. For grouped continuous data: Median = L + [((N/2) - cf) / f] × h (where L is lower boundary of median class, cf is cumulative frequency of preceding class, f is frequency of median class, h is class width). Mode: Most frequent observation. For grouped continuous data: Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × h.

5. Measures of Dispersion: Variance & Standard Deviation

Range = Maximum value - Minimum value. Variance (σ²): Average of squared deviations from mean: σ² = [Σ (xi - x̄)²] / N = [(Σ xi²) / N] - (x̄)². Standard Deviation (σ): Positive square root of variance: σ = √Variance. Coefficient of Variation (CV): Relative dispersion metric: CV = (σ / x̄) × 100. Lower CV indicates higher consistency and reliability.

6. The Empirical Relationship & Mean-Deviation Trick

For moderately skewed frequency distributions, Karl Pearson's empirical formula connects the three central averages: Mode = 3 Median - 2 Mean. This relation allows calculating any third measure in 15 seconds if two are known. Effect of linear transformation on statistics: If every observation is shifted by adding constant k (y = x + k), Mean, Median, and Mode increase by k, but Variance and Standard Deviation remain COMPLETELY UNCHANGED.