If `n_(P_(r-1))/(a)=(n_(P_(r)))/(b)=(n_(P_(r+1)))/(c)`," then which of the following holds good

(n_(P_(r))/(n_(C_(r))))=

For nonnegative integers s and r , let {:((s),(r)):} = {:(s!)/((r! (s-r)!),(" 0"):} {:(" if "r le g),(" if "gts):} For positive integers m and n , let g (m,n) = sum_(p=0)^(m+n) (f(m,n,p))/{:((n+p),("p")):} where for nay nonnegative integer p , f(m,n,p) = sum_(i=0)^(p){:((n+i),(i)),((n+i),(p)),((p+n),(p-i)) Then which of the following statements is/are TRUE ?

If ^(n)P_(r)=^(n)P_(r+1) and ^(n)C_(r)=^(n)C_(r-1, then the ) value of n+r is.

If .^(n)P_(5)+5*^(n)P_(4)=^(10)P_(r), then r=

11. Prove that nP_(r)=n(n-1)P_(r-1)

If .^(n)P_(r)=840,.^(n)C_(r)=35, then find n=

If .^(n)P_(r)=.^(n)P_(r+1) and .^(n)C_(r)=.^(n)C_(r-1) , find n and r.