Principle deep dive
Pascal / binomial-coefficient identities
ΣC(n,r) = 2ⁿ, C(n,r) = C(n,n−r), Pascal's rule. Spans Binomial Theorem and P&C primarily, plus M&D / Sets / Statistics questions where the identity is the key step.
- questions in the bank
- 49
- tagged HARD
- 19%
- chapter spread
- 5
- worked examples below
- 4
When to reach for it
The question involves C(n, r) — a sum, a coefficient in an expansion, a relation in AP/GP, or a selection problem.
Why this principle matters
Four identities cover ~80% of NDA's binomial-coefficient questions. ΣC(n, r) = 2ⁿ (sum of all binomial coefficients). C(n, r) = C(n, n − r) (symmetry). C(n, r) + C(n, r − 1) = C(n + 1, r) (Pascal's rule). And the alternating sum: ΣC(n, r) · (−1)ʳ = 0 — which is the secret to the trickiest Binomial Theorem problems.
The principle reaches into P&C ("how many ways to choose 5 from 8?"), Binomial Distribution (P(X = k) = C(n, k) · pᵏqⁿ⁻ᵏ), Statistics (combinatorial summations), Matrices & Determinants (matrices of binomial coefficients), and Sequence & Series (C(n, r) in AP, etc.). Same identity, different chapter dress.
For 'coefficient of xᵏ' questions, the general term Tᵣ₊₁ = C(n, r) · xⁿ⁻ʳ · yʳ in (x + y)ⁿ is the workhorse. Find which r gives the desired power, plug in. For coefficient relations across two expansions (1 + x)ᵖ(1 + x)ᵠ, combine them into (1 + x)ᵖ⁺ᵠ first — saves time.
4 worked examples from the bank
Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.
Example 1 · Binomial Theorem · Sums of Binomial Coefficients — Alternating, Weighted, and Symmetric
Example 2 · Binomial Theorem · Coefficients and Specific Terms in Expansion
Example 3 · Binomial Theorem · Coefficients and Specific Terms in Expansion
Example 4 · Binomial Theorem · Sums of Binomial Coefficients — Alternating, Weighted, and Symmetric
Variants to recognise
Same principle, different surfaces. Pattern-match these on test day.
Σ C(n, r) = 2ⁿ
Set x = 1 in (1 + x)ⁿ. Variants: Σ even-r = Σ odd-r = 2ⁿ⁻¹ (set x = ±1).
Symmetry: C(n, r) = C(n, n − r)
Symmetric around the middle. Combined with the binomial expansion: 'r-th from end' = 'r-th from start' (with shifted index).
Pascal's rule: C(n, r) = C(n − 1, r − 1) + C(n − 1, r)
Adjacent entries in Pascal's triangle. Used in recursive identities and the rule for C(n+1, r).
General term Tᵣ₊₁ = C(n, r) · xⁿ⁻ʳ · yʳ
The expansion's machinery. Find r so xⁿ⁻ʳyʳ matches the desired power, then evaluate the coefficient.
Drill every pascal / binomial-coefficient identities question
49 questions from the bank — paginated, with cart and Word-export support.
Related principles
Often combined with this one — drill these next if you found the examples above tractable.