If `"^(11)P_r =^12P_(r-1) `, find the value of r.

If .^(n+1)C_(r+1):^(n)C_(r):^(n-1)C_(r-1)=11:6:3 , find the values of n and r.

If a_(n) = sum_(r=0)^(n) (1)/(""^(n)C_(r)) , find the value of sum_(r=0)^(n) (r)/(""^(n)C_(r))

If ^n P_r=^n P_(r+1)a n d^n C_r=^n C_(r-1,) then the value of n+r is.

If Q(0,1)is equidistantfrom P(5,- 3) and R (x, 6),find the values of x.Also find the distances QR and PR.

In a triangle ABC, if r_1 + r_3 + r = r_2 , then find the value of (sec^2 A + cos^2 B-cot^2 C) ,

Find r if "^5P_r = "^6P_(r-1) .

Prove that : ^nP_r= "^(n-1)P_r+r ^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

If |{:(.^(9)C_(4),.^(9)C_(5),.^(10)C_(r)),(.^(10)C_(6 ),.^(10)C_(7),.^(11)C_(r+2)),(.^(11)C_(8),.^(11)C_(9),.^(12)C_(r+4)):}|=0 , then the value of r is equal to

Find r if "^5P_r =2 "^6P_(r-1) .