Question

The value of ²⁰C₁ + 2 × ²⁰C₂ + 3 × ²⁰C₃ + … + 20 × ²⁰C₂₀ is
1. 19 × 20²⁰
2. \(20 \times 2^{19}\)
3. 20 × 220
4. 19 × 2¹⁹

Answer 1

Correct Answer - Option 2 : \(20 \times 2^{19}\)

Concept :

We use expansions of (1 + x)ⁿ and (1 - x)n  for these types of problems.

Expansion: (1 + x)ⁿ = ⁿCn xⁿ + nC_{n - 1} x^{n - 1} + nC_{n - 2} xn - 2 + ...... +ⁿC₀.

Calculation :

If we differentiate this expansion on both L.H.S and R.H.S, we get :

⇒ n × (1 + x)^{n - 1} = n × ⁿC_nx^{n - 1} + (n - 1) × ⁿC_{n - 1}x^{n - 2} +........ + 0.

If we input x = 1 and n = 20 in the above equation, we get :

⇒ 20 × (1 + 1)^{20 - 1} = 20 × ²⁰C₂₀ 1^{20 - 1} + (20 - 1) × ²⁰C₁₉1^{(20 - 2)} + ..... + 0.

⇒ 20 × 2¹⁹ = 20 × ²⁰C₂₀ + 19 × ²⁰C₁₉ + .... + 1 × ²⁰C₁ + 0.

Hence the value of 20C1 + 2 × 20C2 + 3 × 20C3 + … + 20 × 20C20 is 20 × 2¹⁹.

Whenever a question is asked on Binomial coefficients try differentiating or Integrating expansions of \((1+x)^n \) and \((1-x)^n \)