Question

The sum of the last eight coefficients in the expansion of (1 + x)¹⁶ is equal to ______.

Options

• 2¹⁵

• 2¹⁴

• `2^15 - 1/2((16)!)/((8!)^2`

• `2^16 - 161/(81)^2`

Answer 1

The sum of the last eight coefficients in the expansion of (1 + x)¹⁶ is equal to `underlinebb(2^15 - 1/2((16)!)/((8!)^2)`.

Explanation:

(1 + x)¹⁶ = ¹⁶C₀ + ¹⁶C₁x + ¹⁶C₂x² + .... + ¹⁶C₁₆x¹⁶

2¹⁶ = ¹⁶C₀ + ¹⁶C₁ + ¹⁶C₂ +  ..... + ¹⁶C₁₆

⇒ 2¹⁶ = 2(¹⁶C₉ + ¹⁶C₁₀ + ¹⁶C₁₁ + .... + ¹⁶C₁₆) + ¹⁶C₈

⇒ Sum of last eight coefficients = `(2^16 - ""^16C_8)/2`

= `2^15 - underline(|16)/(2(underline(|8))^2`