The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is:

Correct option is C. 48
Let 7 observations be x_1, x_2, x_3, x_4, x_5, x_6, x_7
$\bar{x}$ = 8 ⇒ Σᵢ₌₁⁷xᵢ = 56 (1)
Also σ² = 16 ⇒ 16 = (1/7)(Σᵢ₌₁⁷xᵢ²) - ($\bar{x}$)² ⇒ 16 = (1/7)(Σᵢ₌₁⁷xᵢ²) - 64
(Σᵢ₌₁⁷xᵢ²) = 560 (2)
Now, x₁ = 2, x₂ = 4, x₃ = 10, x₄ = 12, x₅ = 14 ⇒ x₆ + x₇ = 14
x₆² + x₇² = 100
∴ x₆² + x₇² = (x₆ + x₇)² - 2x₆.x₇ ⇒ x₆.x₇ = 48