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 .…

Question

For nonnegative integers s and r , let s r = s ! r ! s - r ! &#160;if&#160; r &#8804; s 0 &#160;if&#160; r &#62; s . For positive integers m and n , let g m , n = &#8721; p = 0 m + n f m , n , p n + p p , where for any nonnegative integer p , f m , n , p = &#8721; i = 0 p m i n + i p p + n p - i . Then which of the following statements is/are TRUE?

Aicharya · Accepted Answer

f m , n , p = &#8721; i = 0 p m i n + i p p + n p - i = &#8721; i = 0 p C i m &#183; n + i ! p ! n + i - p ! &#183; ( p + n ) ! p - i ! n + i ! = &#8721; i = 0 p C i m &#183; ( p + n ) ! p ! n ! &#183; n ! n + i - p ! p - i ! = &#8721; i = 0 p C i m . C p p + n &#183; C p - i n C p p + n &#183; &#8721; i = 0 p C i m &#183; C p - i n &#8757; &#8721; i = 0 p C i m &#183; C p - i n = C p m + n &#8756; f m , n , p = C p p + n . C p m + n &#8658; f m , n , p C p p + n = C p m + n . g m , n = &#8721; p = 0 m + n C p m + n = 2 m + n (A) g ( 2 m , &#160; 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 .

For nonnegative integers s and r , let s r = s ! r ! s - r !  if  r ≤ s 0  if  r…

Solution

For nonnegative integers s and r , let s r = s ! r ! s - r ! &#160;if&#160; r &#8804; s 0 &#160;if&#160; r…

Solution

For nonnegative integers s and r , let s r = s ! r ! s - r ! if r ≤ s 0 if r…