By formula, ⁿc_x = ⁿc_y
⇔ x = y or x + y = n

Now, ⁵⁰c_r = ⁵⁰c_{r + 2}
⇒ r + r + 2 = 50 or r = r + 2

Correct Option: A

By formula, ⁿc_x = ⁿc_y
⇔ x = y or x + y = n

Now, ⁵⁰c_r = ⁵⁰c_{r + 2}
⇒ r + r + 2 = 50 or r = r + 2
⇒ 2r = 48 [∵ r = r + 2 is not possible]
∴ r = 24

⁵ P₂ = 5! / (5 - 2)! = 5 x 4 = 20

Correct Option: C

⁵ P₂ = 5! / (5 - 2)! = 5 x 4 = 20

ⁿ⁺²C₈ : ^n-2P₄ = 57 : 16
⇒ {(n + 2 )! (n - 6)!} / {(n - 6)! (n - 2)! 8!} = 57/16
⇒ (n + 2) (n + 1) n(n - 1) = 143640

Correct Option: B

ⁿ⁺²C₈ : ^n-2P₄ = 57 : 16
⇒ {(n + 2 )! (n - 6)!} / {(n - 6)! (n - 2)! 8!} = 57/16
⇒ (n + 2) (n + 1) n(n - 1) = 143640
⇒ (n² + n - 2) (n² + n ) = 143640
⇒ (n² + n )² - 2(n² + n) + 1 = 143641
⇒ (n² + n - 1)² = (379)²
[∵ n² + n - 1 > 0]
⇒ n² + n - 1 = 379
⇒ n² + n - 380 = 0
⇒ (n + 20) (n - 19) = 0
∴n = 19 ( Since n is not negative)

Required no. of hand shakes = {8 x (8 - 1)}/2 = 28

Correct Option: D

Required no. of hand shakes = {8 x (8 - 1)}/2 = 28

Required number of ways = ²⁵C₃

Correct Option: B

Required number of ways = ²⁵C₃
= (25 x 24 x 23) / (1 x 2 x 3) = 2300