Answer 1

²¹C_6n = `""^21"C"_(("n"^2 + 5))` 

If ⁿC_x = ⁿC_y, then either x = y or x = n – y

∴ 6n = n² + 5 or 6n = 21 – (n² + 5)

∴ n² – 6n + 5 = 0 or 6n = 21 – n² – 5

∴ n² – 6n + 5 = 0 or n² + 6n – 16 = 0

If n² – 6n + 5 = 0 then (n – 1)(n – 5) = 0

∴ n = 1 or n = 5

If n = 5 then

n² + 5 = 30 > 21

∴ n ≠ 5

∴ n = 1

If n² + 6n – 16 = 0 then (n + 8)(n – 2) = 0

n = – 8 or n = 2

n ≠ – 8

∴ n = 2