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MathsMediumClass 11

Multinomial Theorem

Binomial Theorem

9

JEE Qs

8%

Hard

60

min

Master the general term formula and its application to efficiently find specific coefficients, as this is the most common type of problem asked.

🧮 Key Formulas

General term in (x1 + x2 + ... + xk)^n is T = (n! / (n1! n2! ... nk!)) * (x1^n1 * x2^n2 * ... * xk^nk), where n1 + n2 + ... + nk = n and n_i >= 0 are integers.
Coefficient of x1^n1 * x2^n2 * ... * xk^nk in (x1 + x2 + ... + xk)^n is n! / (n1! n2! ... nk!).
Number of distinct terms in the expansion of (x1 + x2 + ... + xk)^n is (n + k - 1) C (k - 1) or equivalently, (n + k - 1) C n.

✅ Key Points for JEE

  • 1The Multinomial Theorem is a generalization of the Binomial Theorem for expanding expressions with more than two terms, like (x1 + x2 + ... + xk)^n.
  • 2The core idea is to find the coefficient of a specific term x1^n1 * x2^n2 * ... * xk^nk by ensuring that the sum of the powers (n1 + n2 + ... + nk) equals the total power 'n'.
  • 3The formula for the coefficient involves factorials of the total power and the individual powers, reflecting the number of ways to arrange 'n' items where n_i items are identical of type i.
  • 4Problems often involve finding a specific coefficient or the sum of coefficients by substituting appropriate values (e.g., 1 for all variables for sum of all coefficients, -1 for some variables).
  • 5The formula for the number of terms is derived from stars and bars method, representing ways to distribute 'n' identical items into 'k' distinct bins.

⚠️ Common Mistakes

  • ✕Incorrectly identifying 'n' (the total power of the multinomial) or 'k' (the number of terms within the parenthesis).
  • ✕Errors in applying the coefficient formula, specifically forgetting to include factorials in the denominator (n1! n2! ... nk!).
  • ✕Failing to ensure that the sum of the powers of the variables in the desired term (n1 + n2 + ... + nk) exactly equals the total power 'n'.

📝 Practice Questions

See all

Q21.If ∑30r=1 r2(30Cr)230Cr−1

2025·NumericalMedium

Q20.If the area of the region {(x, y) : −1 ≤x ≤1, 0 ≤y ≤a + e|x| −e−x, a > 0} is e2+8e+1e , then the value of is : (1) 8 (2) 7 (3) 5 (4) 6

2025·MCQMedium

Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1

2025·NumericalHard

Q3. Let α, β, γ and δ be the coefficients of x7, x5, x3 and x respectively in the expansion of 5 5 αu + βv = 18 + , x > 1. If u and v satisfy the equations then u + v equals : (x + √x3 −1) (x −√x3 −1) γu + δv = 20 (1) 5 (2) 3 (3) 4 (4) 8

2025·MCQHard

Q24.The sum of all rational terms in the expansion of (1 + 21/2 + 31/2) 6 is equal to

2025·NumericalMedium

Q6. The product of all the rational roots of the equation (x2 −9x + 11)2 −(x −4)(x −5) = 3, is equal to (1) 14 (2) 21 (3) 28 (4) 7

2025·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 7: Permutations and Combinations
  • Class 11 Maths Ch 8: Binomial Theorem