Question

The value of the sum of the series

¹⁴C₀ · ¹⁵C₁ + ¹⁴C₁ · ¹⁵C₂ + ¹⁴C₂ · ¹⁵C₃ +…. + ¹⁴C₁₄ · ¹⁵C₁₅ is –

• A. ²⁹C₁₂
• B. ²⁹C₁₀
• C. ²⁹C₁₄
• D. ²⁹C₁₆

Answer 1

Answer: (C)

²⁹C₁₄

Answer 2

We have,

(1 + x)¹⁴ = ¹⁴C₀ + ¹⁴C₁x + ¹⁴C₂x² +…..+ ¹⁴C₁₄ x¹⁴ …(1)

and (x + 1)¹⁵ = ¹⁴C₀x¹⁵ + ¹⁵C₁x ¹⁴ + ¹⁵C₂x¹³ + ¹⁵C₃x¹²…..+ ¹⁵C₁₅ …(2)

Multiplying (1) and (2) and equating the co-efficient of x¹⁴, we get

¹⁴C₀ · ¹⁵C₁ + ¹⁴C₁ · ¹⁵C₂ + ¹⁴C₂ · ¹⁵C₃ + ….. + ¹⁴C₁₄ · ¹⁵C₁₅

= the coefficient of x¹⁴ in (1 + x)²⁹ = ²⁹C₁₄.