If t_n denotes the number of triangles formed with n points in a plane, no three of which are collinear and if t_ n+1 -t_n=36 , then n is equal to

Step-by-step solution

t_n= The number of triangles formed with n points in a plane, no three of which are collinear. i.e., aligned & t_n= ^n C₃ & t_ n+1 = ^ n+1 C₃ aligned t_ n+1 = ^ n+1 C₃ Now, t_ n+1 -t_n=36 array crrl & ^ n+1 C₃- ^n C₃=36 & (n+1) ! (n-2) ! 3 ! - n ! (n-3) ! 3 ! =36 & (n+1) n(n-1) 6 - n(n-1)(n-2) 6 =36 & & n(n-1)(n+1-n+2)=36 6 & & 3 n(n-1)=36 6 & & n^2-n-72=0 & & n^2-9 n+8 n-72=0 & & n(n-9)+8(n-9)=0 & & (n-9)(n+8)=0 & & n=9 ( n -3) array