If the number of terms in the expansion of (x 180 + [3] 432 )²⁰⁰ having integral coefficients is n , then the vaule of [ n 6 ] is (where [.] denotes greater integer function)

Step-by-step solution

aligned & T_ r+1 = ²⁰⁰ C_r(x 180 )^ 200-r ( [3] 432 )^r & = ²⁰⁰ C_r 2^ 200-r 3^ 200-r 5^ 200-r 2 x^ 200-r 2^r 3^r 2^ r 3 & Coefficient = ²⁰⁰ C_r 2²⁰⁰ 3²⁰⁰ 5¹⁰⁰ 2^ r 3 5^ r 2 aligned r must be multiple of 6 and takes values between 0 to 200. 34 terms =n [ n 6 ]= [ 34 6 ]=5