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🧪 Chemistry — Major

Unit VIII — Mathematical Concepts for Chemistry · Calculus & Probability for Chemists

Permutations, Combinations & Probability

CompleteFundamentals of Chemistry

Counting arrangements and Likelihood — the basis of statistical mechanics and molecular velocities.

Full notes

1. Factorials

n! = n(n−1)(n−2)…1. 0! = 1. Factor 12! = 479001600 — huge numbers arise naturally in statistical counting.

2. Permutations (ordered arrangements)

ⁿPᵣ = n!/(n−r)! counts ordered selections of r objects from n. Example: arranging 3 electrons in 2 states.

3. Combinations (unordered selections)

ⁿCᵣ = n!/(r!(n−r)!) counts unordered subsets. Example: choosing 2 of 3 reactants to react.

4. Probability

P(event) = favourable outcomes / total outcomes. Independent events multiply; mutually exclusive events add. In chemistry: probability of finding an electron in a region relates to ψ²·dτ; distribution of molecular speeds is a probability distribution (Maxwell-Boltzmann).

Example: ⁿCₙ=1, ⁿC₁=n, ⁿCᵣ = ⁿCₙ₋ᵣ. For a molecule with two energy levels, the probability of finding it in the lower level at equilibrium follows the Boltzmann factor e^(−ΔE/kBT).
Note: Exam point: ⁿPᵣ orders matter; ⁿCᵣ orders don't. Toss a coin twice → four equally likely outcomes, so P(HH) = 1/4.

Exam notes

ⁿPᵣ=n!/(n−r)!; ⁿCᵣ=n!/(r!(n−r)!). Probability = fav/total. These appear in statistics/steric counting and quantum states.

Important points

  • The permutation/combination and probability concepts recur in stereoisomer counting and statistical thermodynamics.

Common mistakes

  • Using permutation when selection order is irrelevant.
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