Binomial Theorem
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Pascal Triangle and Expansion
Part of: Expansion and Terms
Pascal rows add adjacent pairs; index-n rows hold n-plus-one entries; (a+b)^n sums nCr a-powers b-powers with exponents totalling n.
Why it matters. Rows read coefficients instantly. Exponent sums check terms.
Formulas
- (a + b)^n = sum of nCr a^(n-r) b^r for r = 0 to n.
- (a + b)^n has exactly n + 1 terms, one more than the index.
- Exponents of a and b in each term add to n.
Key words
- Binomial theorem
- : The rule expanding (a + b)^n into nCr-weighted terms.
- Pascal triangle
- : The triangle where each entry adds the two above it.
- Term count
- : The count n + 1 of terms in the expansion of index n.
- Exponent pattern
- : The falling powers of a matched by rising powers of b.
Try it in two minutes
- Build the Pascal row.
- Attach falling a-powers.
- Check exponent sums.
Try at home. Write row 6 of Pascal.
General Term and Middle Terms
Part of: Expansion and Terms
Term r-plus-one equals nCr a-to-n-minus-r b-to-r; odd index splits into two middles; even index centres one; independent terms zero the x-exponent.
Why it matters. Direct terms skip expansion. Middle terms anchor symmetry.
Formulas
- General term: T(r+1) = nCr a^(n-r) b^r.
- If n is even, the middle term is the (n/2 + 1)th term.
- If n is odd, the two middle terms are the ((n+1)/2)th and ((n+3)/2)th.
Key words
- General term
- : The formula T(r+1) that picks any single term without full expansion.
- Middle term
- : The central term of an even-index expansion, found by halving n.
- Independent term
- : Setting the x-exponent to zero to find the term free of x.
- Term index
- : The r inside nCr that positions the term within the row.
Try it in two minutes
- Index from zero.
- Set exponents.
- Solve for the slot.
Try at home. Find term 3 of (x + 2)^6.
Properties of Binomial Coefficients
Part of: Coefficients and Uses
Coefficients sum to two-to-n; mirror pairs match; even index peaks once, odd peaks twice; Pascal recurrence adds upper neighbours.
Why it matters. Sums count subsets. Peaks spot largest terms.
Formulas
- C(n, r) = C(n, n - r), so mirror coefficients match.
- C(n, 0) + C(n, 1) + ... + C(n, n) = 2^n.
- C(n, r) + C(n, r - 1) = C(n + 1, r), the Pascal addition rule.
Key words
- Binomial coefficient
- : The nCr numbers weighting each term of the expansion.
- Symmetry
- : The mirror equality C(n, r) = C(n, n - r) across a row.
- Coefficient sum
- : The row total 2^n found by setting x = 1 in (1 + x)^n.
- Maximum coefficient
- : The middle peak of coefficients, single for even n and double for odd n.
Try it in two minutes
- Sum to two-to-n.
- Mirror the pairs.
- Find the peak.
Try at home. Sum row 5 of Pascal.
Approximations and Divisibility
Part of: Coefficients and Uses
Neglecting high powers approximates decimals; splitting bases reveals factors; expanded remainders prove divisibility.
Why it matters. First terms estimate fast. Expansions prove divisibility.
Formulas
- (x - y)^n alternates signs, ending with (-1)^n times the last term.
- (1 + x)^n = nC0 + nC1 x + nC2 x^2 + ... + nCn x^n.
- (1 - x)^n = nC0 - nC1 x + nC2 x^2 - ... with alternating signs.
Key words
- Approximation
- : Estimating a power by keeping only the first few expansion terms.
- Base splitting
- : Splitting a base like 98 into 100 - 2 to expose easy powers.
- Divisibility proof
- : Showing a remainder is zero by expanding and grouping factors.
- Neglected terms
- : The tiny leftover terms dropped when approximating decimals.
Try it in two minutes
- Split the base.
- Keep first terms.
- Factor the rest.
Try at home. Estimate 1.02^10 roughly.
What your child will practise
- Approximations and Divisibility: Space Puzzle (Clue Puzzle)
- Approximations and Divisibility: Space Quick Fire (Quick Fire)
- General Term and Middle Terms: Ocean Puzzle (Clue Puzzle)
- Pascal Triangle and Expansion: Jungle Quest (Flip and Match)
- Properties of Binomial Coefficients: Candy Quick Fire (Quick Fire)
- Quiz: Approximations and Divisibility (8 questions)
- Quiz: General Term and Middle Terms (8 questions)
- Quiz: Pascal Triangle and Expansion (8 questions)
- Quiz: Properties of Binomial Coefficients (8 questions)