For nonnegative integers s and r , let s r = s ! r ! s - r !  if  r ≤ s 0  if  r > s . For positive integers m and n , let g m , n = ∑ p = 0 m + n f m , n , p n + p p , where for any nonnegative integer p , f m , n , p = ∑ i = 0 p m i n + i p p + n p - i . Then which of the following statements is/are TRUE?
D. g 2 m ,   2 n = g m , n 2 for all positive intergers m ,   n
f m , n , p = ∑ i = 0 p m i n + i p p + n p - i = ∑ i = 0 p C i m · n + i ! p ! n + i - p ! · ( p + n ) ! p - i ! n + i ! = ∑ i = 0 p C i m · ( p + n ) ! p ! n ! · n ! n + i - p ! p - i ! = ∑ i = 0 p C i m . C p p + n · C p - i n C p p + n · ∑ i = 0 p C i m · C p - i n ∵ ∑ i = 0 p C i m · C p - i n = C p m + n ∴ f m , n , p = C p p + n . C p m + n ⇒ f m , n , p C p p + n = C p m + n . g m , n = ∑ p = 0 m + n C p m + n = 2 m + n (A) g ( 2 m ,   2 n ) = 2 2 m + 2 n 2 g m , n = 2 . 2 m + n (B) g m , n + 1 = 2 m + n + 1 = g m + 1 , n (C) g 2 m , 2 n = 2 2 m + 2 n = 2 m + n 2 = g m , n 2 (D) g m , n = g n , m .
Related: Mathematics — Binomial Theorem · All PYQ Banks