If `10C_r=^10C_2` find the values of r.

If .^(n)C_(r )=84, .^(n)C_(r-1)=36 " and" .^(n)C_(r+1)=126 , then find the value of n.

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

When n = 5 and r = 2, find the values of : (n!)/(r!) .

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) ,

Let n be a positive integer and Delta_(r)=|{:(r^(2)+r,r+1,r-2),(2r^(2)+3r-1,3r,3r-3),(r^(2)+2r+3,2r-1,2r-1):}| and sum _(r=1)^(n)Delta_(r)=an^(2)+bn+c find the value of a+b+c.

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

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

If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (- 4, 3b,-10) and R(8, 14, 2c), then find the values of a, b and c.

It is desired to make a 20.0Omega coil of wire whose temperature coefficient of resistance is zero. To do this, a carbon resistor of resistance R_(1) is placed in series with an iron resistor of resistance R_(2) . The proportion of iron and carbon are so chosen that R_(1) +R_(2)= 20 Omega for all temperatures near 20^@C . Find the values of R_(1) and R_(2). Temperature coefficient of resistance for carbon, alpha_(C)= - 0.5 xx 10^(-3)//^(@)C and that of iron is alpha_(Fe) =5 xx 10^(-3)//^(@)C.