Let 0 ≤ r ≤ n. If ⁿ⁺¹ Cᵣ₊₁: ⁿ Cᵣ: ⁿ⁻¹ Cᵣ₋₁=55: 35: 21, then 2 n+5 r is equal to :

Solution

Maths sir · 4 steps

Do not expand all three: link each coefficient to the next one and the ratios hand you n and r.

1. Useful identity. From the factorial formula, ⁿCᵣ = n/r ⁿ⁻¹Cᵣ₋₁. Each coefficient is a simple multiple of the one below it, so each ratio of neighbours becomes a plain fraction.

   ⁿ⁺¹Cᵣ₊₁/ⁿCᵣ = (n+1)/(r+1) = 55/35, ⁿCᵣ/ⁿ⁻¹Cᵣ₋₁ = n/r = 35/21

2. Two linear equations. (n+1)/(r+1) = 11/7 gives 7n - 11r = 4. n/r = 5/3 gives 3n = 5r, so n = 5r/3.
3. Solve. Substituting, 35r/3 - 11r = 4, so 2r/3 = 4 and r = 6, n = 10. Check: ¹¹C₇ : ¹⁰C₆ : ⁹C₅ = 330 : 210 : 126 = 55 : 35 : 21.
4. So. 2n + 5r = 20 + 30 = 50, which is option (d).

The idea to keep. When consecutive binomial coefficients are given in a ratio, use ⁿCᵣ = n/r ⁿ⁻¹Cᵣ₋₁. Each ratio turns into a linear equation in n and r.

Answer from the NTA final answer key. Solution by PrepWiser.