In the binomial expansion of (x^ + c x^ )¹⁰ , where and are positive integers and c > 0 , the ratio of the 3^ rd term from the end to the 3^ rd term from the beginning is given as 64x⁻²⁴ . The value of the term independent of x in the expansion is

Step-by-step solution

Let the given binomial be (a+b)^n , where a = x^ , b = cx^ - , and n = 10 . The k^ th term from the end is the (n-k+2)^ th term from the beginning. For k=3 , the 3^ rd term from the end is T_ 10-3+2 = T₉ . The 3^ rd term from the beginning is T₃ . The ratio of the 3^ rd term from the end to the 3^ rd term from the beginning is: T₉ T₃ = ¹⁰C₈ (x^ )^2 (cx^ - )^8 ¹⁰C₂ (x^ )^8 (cx^ - )^2 Since ¹⁰C₈ = ¹⁰C₂ , this simplifies to: (cx^ - )^6 (x^ )^6 = c^6 x^ -6( + ) We are given that this ratio is 64x⁻²⁴ . Comparing the two