Meaning of Binomial Coefficients and Pascal's Triangle
Simple Explanation
A binomial coefficient, written ⁿCᵣ (read "n choose r"), counts how many ways you can choose r items from a group of n, where order doesn't matter. These same numbers appear as the coefficients when expanding (a+b)ⁿ, and can be read directly off row n of Pascal's triangle — a triangular array where each number is the sum of the two numbers above it.
Why Do We Need It?
Binomial coefficients are the bridge between counting (combinations) and algebra (binomial expansions) — the same numbers do both jobs, which is exactly why the binomial theorem works.
Formula
The Combination Formula (Binomial Coefficient)
ⁿCᵣ = n! / (r!(n − r)!)
Counts how many ways to choose r items from a set of n, when order does not matter — this is exactly the coefficient of each term in a binomial expansion.
- n
- — the total number of items (or the power in the binomial expansion)
- r
- — the number of items chosen (or the term index in the expansion)
- n!
- — n factorial: n × (n−1) × (n−2) × ... × 1
When to use it: Whenever you need a binomial coefficient — either for counting combinations or for the coefficients in a binomial expansion.
Worked Example
Evaluate a binomial coefficient
Evaluate ⁵C₂.
Why Does This Work?
Pascal's triangle's "each number is the sum of the two above it" rule directly matches the identity ⁿCᵣ = ⁿ⁻¹Cᵣ₋₁ + ⁿ⁻¹Cᵣ (choosing r items either includes a specific item, using ⁿ⁻¹Cᵣ₋₁ ways to pick the rest, or excludes it, using ⁿ⁻¹Cᵣ ways) — so building the triangle row by row always reproduces the exact same numbers as the combination formula.
Real-Life Example
Choosing a committee
A club with 10 members wants to know how many different 3-person committees it could form.
¹⁰C₃ gives that count directly — order doesn't matter for a committee, exactly the situation binomial coefficients are built for.
Practice
Evaluate ⁶C₂.
MediumCommon mistake
Confusing ⁿCᵣ (order doesn't matter) with ⁿPᵣ (order matters, permutations) — binomial coefficients always use combinations, never permutations.
Quick Review
- ⁿCᵣ = n! / (r!(n−r)!) — the number of ways to choose r from n, order not mattering.
- These are exactly the coefficients in the expansion of (a+b)ⁿ.
- Row n of Pascal's triangle lists ⁿC₀, ⁿC₁, ..., ⁿCₙ in order.