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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
48
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 1Binomial TheoremEASY
If C0,C1,C2,…,CnC_0, C_1, C_2, \ldots, C_n are the coefficients in the expansion of (1+x)n(1+x)^n, then what is the value of C1+C2+C3+⋯+CnC_1 + C_2 + C_3 + \cdots + C_n?

[Q4 · Apr · 2021]

Example 2Binomial TheoremEASY
In the expansion of (1+x)p(1+x)q(1+x)^{p}(1+x)^{q}, if the coefficient of x3x^{3} is 35, then what is the value of (p+q)(p+q)?

[Q6 · Sep · 2024]

Example 3Binomial TheoremEASY
Consider the following statements in respect of the expansion of (x+y)10(x+y)^{10}: 1. Among all the coefficients of the terms, the coefficient of the 6th term has the highest value. 2. The coefficient of the 3rd term is equal to coefficient of the 9th term. Which of the above statements is/are correct?

[Q12 · Apr · 2022]

Example 4Binomial TheoremHARD
What is the value of C(51,21)−C(51,22)+C(51,23)−C(51,24)+C(51,25)−C(51,26)+C(51,27)−C(51,28)+C(51,29)−C(51,30)C(51,21)-C(51,22)+C(51,23)-C(51,24)+C(51,25)-C(51,26)+C(51,27)-C(51,28)+C(51,29)-C(51,30)?

[Q14 · Apr · 2022]

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

48 questions from the bank — paginated, with cart and Word-export support.

Drill the 48 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.