The coefficient of the term independent of x in the expansion of ( x 3 + 3 2 x² )¹⁰ is

Step-by-step solution

The (r+1) th term in the expansion of ( x 3 + 3 2 x² )¹⁰ is given by array l T _ r+1 = ¹⁰ C _ r ( x 3 )^ 10-r ( 3 2 x² )^ r = ¹⁰ C _ r x^ 5- ( r 2 ) 3^ 5- ( r 2 ) 3^ r 2^ r x^ 2 r = ¹⁰ C _ r 3^ ( 3 r 2 )-5 -5- ( 5 r 2 ) 2^ r x array T _ r+1 = ¹⁰ C _ r ( x 3 )^ 10-r ( 3 2 x² )^ r = ¹⁰ C _ r x^ 5- ( r 2 ) 3 3^ r 2^ r x^ 2 r = ¹⁰ C _ r 3^ ( 3 r 2 )-5 - ( 5 2 )- ( 5 r 2 ) 2^ r x For T _ r+1 to be independent of x , we must have 5- ( 5 r 2 )=0 or r=2 . Thus, the 3 rd term is independent of x and is equal to ¹⁰ C ₂ 3³⁻⁵