If the coefficients of x^9, x¹⁰ and x¹¹ in the expansion of (1+x)^n are in arithmetic progression, then n^2-41 n is equal to

Step-by-step solution

Given that coefficient of x^9, x¹⁰ and x¹¹ in the expansion of (1+n)^n are in A.P. It means ^n C₉ ^n C₁₀ ^n C₁₁ are in A.P. aligned & 2^n C₁₀= ^n C₉+ ^n C₁₁ & 2= ^n C₉ ^n C₁₀ + ^n C₁₁ ^n C₁₀ & 2= n ! 9 !(n-9) ! 10 !(n-10) ! n ! aligned + n ! 11 !(n-11) ! 10 !(n-10) ! n ! aligned & 2= 10 n-9 + n-10 11 & 2= 110+(n-10)(n-9) 11(n-9) & 2= 110+n^2-9 n-10 n+90 11 n-99 & 22 n-198=200+n^2-19 n & n^2-41 n=-398 aligned