The ratio of the coefficient of x¹⁵ to the term independent of x in the expansion of (x^2+ 2 x )¹⁵ is:

Step-by-step solution

aligned & T _ r+1 = ¹⁵ C _r (x^2 )^ 15-r (2 x⁻¹ )^r & = ¹⁵ C _r (2)^r x^ 30-3 r aligned For independent term, 30-3 r=0 r=10 Hence the term independent of x , T ₁₁= ¹⁵ C ₁₀ (2)¹⁰ For term involve x¹⁵, 30-3 r=15 r=5 Hence coefficient of x¹⁵= ¹⁵ C ₅ (2)^5 Required ratio aligned & = ¹⁵ C ₅ (2)^5 ¹⁵ C ₁₀ (2)¹⁰ = 15 ! 10 ! 5 ! 15 ! 5 ! 10 ! (2)^5 & =1: 32 & aligned