JEE Main 2025 · 24 January · evening · Single correct · +4 / −1
Suppose A and B are the coefficients of 30ᵗʰ and 12ᵗʰ terms respectively in the binomial expansion of (1+x)²ⁿ⁻¹. If 2 A=5 B, then n is equal to:
- (a) 20
- (b) 19
- (c) 22
- (d) 21
Answer and solution
Answer (d) 21
Solution
Maths sir · 4 steps
Use C(m, r)=C(m, m-r) to flip the 30th coefficient until it sits right beside the 12th.
- Write A and B. Let m = 2n-1. The (r+1)th term of (1+x)ᵐ has coefficient C(m, r). So A = C(m, 29) and B = C(m, 11), and 2A = 5B means A/B = 5/2.
- Flip A. By symmetry, A = C(m, 29) = C(m, m-29). If m - 29 = 12, then A = C(m, 12) is the coefficient right after B = C(m, 11). The ratio of neighbouring coefficients is (m-r)/(r+1).
A/B = C(m, 12)/C(m, 11) = (m-11)/12 = 5/2 ⇒ m = 41
- Check consistency. m = 41 gives m - 29 = 12, which matches the assumption. Now 2n - 1 = 41, so n = 21.
n m = 2n-1 A rewritten A/B 20 39 C(39, 10) 11/29 19 37 C(37, 8) (9 × 10 × 11)/(27 × 28 × 29), far below 1 22 43 C(43, 14) (30 × 31 × 32)/(12 × 13 × 14) ≈ 13.6 21 41 C(41, 12) 30/12 = 5/2 - So. Only n = 21 gives A/B = 5/2, so the answer is (d).
The idea to keep. The (r+1)th term carries C(m, r). When two coefficients look far apart, flip one with C(m, r)=C(m, m-r). Neighbouring coefficients then have the easy ratio (m-r)/(r+1).
Answer from the NTA final answer key. Solution by PrepWiser.